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class="vp-sidebar-children"><!--[--><li><a class="auto-link external-link vp-sidebar-item" href="https://github.com/thejinchao/turbolink" aria-label="TurboLink" rel="noopener noreferrer" target="_blank"><!--[--><!--[--><!--]--><!--]-->TurboLink<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item vp-sidebar-heading active collapsible">学习笔记 <span class="down arrow"></span></p><ul style="" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item active collapsible">数学相关 <span class="down arrow"></span></p><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/symbol.html" aria-label="常用数学符号"><!--[--><!--[--><!--]--><!--]-->常用数学符号<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/group.html" aria-label="群"><!--[--><!--[--><!--]--><!--]-->群<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_1.html" aria-label="数论（一）"><!--[--><!--[--><!--]--><!--]-->数论（一）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_2.html" aria-label="数论（二）"><!--[--><!--[--><!--]--><!--]-->数论（二）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_3.html" aria-label="数论（三）"><!--[--><!--[--><!--]--><!--]-->数论（三）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link route-link-active auto-link vp-sidebar-item active" href="/note/math/probability.html" aria-label="概率"><!--[--><!--[--><!--]--><!--]-->概率<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_1-随机变量" aria-label="1.随机变量"><!--[--><!--[--><!--]--><!--]-->1.随机变量<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_1-1-离散随机变量" aria-label="1.1 离散随机变量"><!--[--><!--[--><!--]--><!--]-->1.1 离散随机变量<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_1-2-连续随机变量" aria-label="1.2 连续随机变量"><!--[--><!--[--><!--]--><!--]-->1.2 连续随机变量<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="#_2-概率分布函数-cdf" aria-label="2. 概率分布函数(CDF)"><!--[--><!--[--><!--]--><!--]-->2. 概率分布函数(CDF)<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_2-1-定义" aria-label="2.1 定义"><!--[--><!--[--><!--]--><!--]-->2.1 定义<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_2-2-举例" aria-label="2.2 举例"><!--[--><!--[--><!--]--><!--]-->2.2 举例<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_2-3-特性" aria-label="2.3 特性"><!--[--><!--[--><!--]--><!--]-->2.3 特性<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-概率密度函数-pdf" aria-label="3. 概率密度函数(PDF)"><!--[--><!--[--><!--]--><!--]-->3. 概率密度函数(PDF)<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_3-1-定义" aria-label="3.1 定义"><!--[--><!--[--><!--]--><!--]-->3.1 定义<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-2-特性" aria-label="3.2 特性"><!--[--><!--[--><!--]--><!--]-->3.2 特性<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-3-举例" aria-label="3.3 举例"><!--[--><!--[--><!--]--><!--]-->3.3 举例<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="#_4-期望值" aria-label="4. 期望值"><!--[--><!--[--><!--]--><!--]-->4. 期望值<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_4-1-定义" aria-label="4.1 定义"><!--[--><!--[--><!--]--><!--]-->4.1 定义<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_4-2-特性" aria-label="4.2 特性"><!--[--><!--[--><!--]--><!--]-->4.2 特性<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="#_5-大数定理" aria-label="5. 大数定理"><!--[--><!--[--><!--]--><!--]-->5. 大数定理<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_6-蒙特卡洛积分法" aria-label="6. 蒙特卡洛积分法"><!--[--><!--[--><!--]--><!--]-->6. 蒙特卡洛积分法<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_6-1-定义" aria-label="6.1 定义"><!--[--><!--[--><!--]--><!--]-->6.1 定义<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_6-2举例" aria-label="6.2举例"><!--[--><!--[--><!--]--><!--]-->6.2举例<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="#_7-采样函数" aria-label="7. 采样函数"><!--[--><!--[--><!--]--><!--]-->7. 采样函数<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_7-1-证明" aria-label="7.1 证明"><!--[--><!--[--><!--]--><!--]-->7.1 证明<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_7-2-举例" aria-label="7.2 举例"><!--[--><!--[--><!--]--><!--]-->7.2 举例<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">密码学 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/rsa.html" aria-label="RSA"><!--[--><!--[--><!--]--><!--]-->RSA<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/flip_coin.html" aria-label="抛币协议"><!--[--><!--[--><!--]--><!--]-->抛币协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/mental_poker.html" aria-label="智能扑克协议"><!--[--><!--[--><!--]--><!--]-->智能扑克协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">图形学 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">数学基础 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_01.html" aria-label="矢量"><!--[--><!--[--><!--]--><!--]-->矢量<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_02.html" aria-label="矩阵"><!--[--><!--[--><!--]--><!--]-->矩阵<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_03.html" aria-label="立体角"><!--[--><!--[--><!--]--><!--]-->立体角<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_01.html" aria-label="几何变换(一)"><!--[--><!--[--><!--]--><!--]-->几何变换(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_02.html" aria-label="几何变换(二)"><!--[--><!--[--><!--]--><!--]-->几何变换(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_03.html" aria-label="法线变换"><!--[--><!--[--><!--]--><!--]-->法线变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_04.html" aria-label="摄像机变换"><!--[--><!--[--><!--]--><!--]-->摄像机变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">光照模型 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/classic/illumination_model_01.html" aria-label="传统光照模型"><!--[--><!--[--><!--]--><!--]-->传统光照模型<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/luminosity/luminosity.html" aria-label="光度学"><!--[--><!--[--><!--]--><!--]-->光度学<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/BRDF/BRDF.html" aria-label="双向反射分布函数(BRDF)"><!--[--><!--[--><!--]--><!--]-->双向反射分布函数(BRDF)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_1.html" aria-label="微平面理论(一)"><!--[--><!--[--><!--]--><!--]-->微平面理论(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_2.html" aria-label="微平面理论(二)"><!--[--><!--[--><!--]--><!--]-->微平面理论(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_3.html" aria-label="微平面理论(三)"><!--[--><!--[--><!--]--><!--]-->微平面理论(三)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/render_function/render_function_1.html" aria-label="光照方程"><!--[--><!--[--><!--]--><!--]-->光照方程<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">环境光渲染 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_01.html" aria-label="环境光渲染(一)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_02.html" aria-label="环境光渲染(二)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">编程语言 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">JavaScript <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/helloworld.html" aria-label="环境搭建"><!--[--><!--[--><!--]--><!--]-->环境搭建<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/grammar.html" aria-label="基本语法"><!--[--><!--[--><!--]--><!--]-->基本语法<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/function.html" aria-label="函数"><!--[--><!--[--><!--]--><!--]-->函数<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/object_and_class.html" aria-label="对象和类"><!--[--><!--[--><!--]--><!--]-->对象和类<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul><!--[--><!--]--></aside><!--]--><!--[--><main class="vp-page"><!--[--><!--]--><div vp-content><!--[--><!--]--><div><h1 id="概率" tabindex="-1"><a class="header-anchor" href="#概率"><span>概率</span></a></h1><hr><h2 id="_1-随机变量" tabindex="-1"><a class="header-anchor" href="#_1-随机变量"><span>1.随机变量</span></a></h2><h3 id="_1-1-离散随机变量" tabindex="-1"><a class="header-anchor" href="#_1-1-离散随机变量"><span>1.1 离散随机变量</span></a></h3><p>比如扔一个质地均匀的色子，产生的结果就是一个离散随机变量，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo fence="false" stretchy="false">}</mo></math></mjx-assistive-mml></mjx-container>，离散随机变量中的每一个可能的值<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container>都对应一个概率<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container>，对于扔色子这个例子，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>i</mi></msub><mo>=</mo><mn>1</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>6</mn></math></mjx-assistive-mml></mjx-container>,所有离散随机数值<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container>的概率之和为 <mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-sop"><mjx-c class="mjx-c2211 TEX-S1"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><msub><mi>p</mi><mi>i</mi></msub></mrow><mo>=</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container></p><h3 id="_1-2-连续随机变量" tabindex="-1"><a class="header-anchor" href="#_1-2-连续随机变量"><span>1.2 连续随机变量</span></a></h3><p>计算机中产生一个[0,1]之间的随机函数rand()，在不考虑浮点数精度时，就是一个连续的随机变量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></mjx-assistive-mml></mjx-container></p><h2 id="_2-概率分布函数-cdf" tabindex="-1"><a class="header-anchor" href="#_2-概率分布函数-cdf"><span>2. 概率分布函数(CDF)</span></a></h2><h3 id="_2-1-定义" tabindex="-1"><a class="header-anchor" href="#_2-1-定义"><span>2.1 定义</span></a></h3><p>衡量一个随机数的概率分布规律的两个重要函数，概率分布函数(cumulative distribution function)用来定义概率累计的结果</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D443 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2A7D TEX-A"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>⩽</mo><mi>x</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><p>其中函数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D443 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math></mjx-assistive-mml></mjx-container>表示某事件发生的概率，也就是说<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>表示随机变量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></mjx-assistive-mml></mjx-container>小于<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></mjx-assistive-mml></mjx-container>的概率</p><h3 id="_2-2-举例" tabindex="-1"><a class="header-anchor" href="#_2-2-举例"><span>2.2 举例</span></a></h3><p>比如对于一个在[a,b]之间概率均匀分布的随机变量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></mjx-assistive-mml></mjx-container></p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:4.427em;vertical-align:-1.964em;" class="mjx-c7B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-mid><mjx-c></mjx-c></mjx-mid><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:7.961em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-bottom:0.1em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-bottom:0.1em;"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-top:0.1em;padding-bottom:0.1em;"><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-top:0.1em;padding-bottom:0.1em;"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2A7D TEX-A"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-top:0.1em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-top:0.1em;"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2A7D TEX-A"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n" style="vertical-align:0.25em;"></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable columnalign="left left" columnspacing="1em" rowspacing=".2em"><mtr><mtd><mn>0</mn></mtd><mtd><mrow data-mjx-texclass="ORD"><mi>x</mi><mo><</mo><mi>a</mi></mrow></mtd></mtr><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>x</mi><mo>−</mo><mi>a</mi></mrow><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac></mstyle></mtd><mtd><mrow data-mjx-texclass="ORD"><mi>a</mi><mo>⩽</mo><mi>x</mi><mo><</mo><mi>b</mi></mrow></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow data-mjx-texclass="ORD"><mi>b</mi><mo>⩽</mo><mi>x</mi></mrow></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo></mrow></math></mjx-assistive-mml></mjx-container><p>特别的，一个在[0,1]之间均匀分布的随机变量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D448 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math></mjx-assistive-mml></mjx-container>的概率分布函数</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D448 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.4em;vertical-align:-1.45em;" class="mjx-c7B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-mid><mjx-c></mjx-c></mjx-mid><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:5.811em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-bottom:0.1em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-bottom:0.1em;"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-top:0.1em;padding-bottom:0.1em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-top:0.1em;padding-bottom:0.1em;"><mjx-TeXAtom texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2A7D TEX-A"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-top:0.1em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-top:0.1em;"><mjx-TeXAtom texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2A7D TEX-A"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n" style="vertical-align:0.25em;"></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mi>U</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable columnalign="left left" columnspacing="1em" rowspacing=".2em"><mtr><mtd><mn>0</mn></mtd><mtd><mrow data-mjx-texclass="ORD"><mi>x</mi><mo><</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mi>x</mi></mtd><mtd><mrow data-mjx-texclass="ORD"><mn>0</mn><mo>⩽</mo><mi>x</mi><mo><</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mrow data-mjx-texclass="ORD"><mn>1</mn><mo>⩽</mo><mi>x</mi></mrow></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo></mrow></math></mjx-assistive-mml></mjx-container><h3 id="_2-3-特性" tabindex="-1"><a class="header-anchor" href="#_2-3-特性"><span>2.3 特性</span></a></h3><ul><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2264"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2264"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mn>1</mn></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>为单调上升的右连续函数</li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mstyle><mjx-munder><mjx-row><mjx-base style="padding-left:0.49em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6D"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.188em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c221E"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder><mjx-TeXAtom space="2" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-munder space="2"><mjx-row><mjx-base style="padding-left:0.49em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6D"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.188em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c221E"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder><mjx-TeXAtom space="2" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><munder><mo data-mjx-texclass="OP" movablelimits="true">lim</mo><mrow data-mjx-texclass="ORD"><mi>x</mi><mo accent="false" stretchy="false">→</mo><mo>−</mo><mi mathvariant="normal">∞</mi></mrow></munder><mrow data-mjx-texclass="ORD"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><mo>,</mo><munder><mo data-mjx-texclass="OP" movablelimits="true">lim</mo><mrow data-mjx-texclass="ORD"><mi>x</mi><mo accent="false" stretchy="false">→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munder><mrow data-mjx-texclass="ORD"><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow></mstyle></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D443 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2A7D TEX-A"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>a</mi><mo><</mo><mi>x</mi><mo>⩽</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></li></ul><h2 id="_3-概率密度函数-pdf" tabindex="-1"><a class="header-anchor" href="#_3-概率密度函数-pdf"><span>3. 概率密度函数(PDF)</span></a></h2><h3 id="_3-1-定义" tabindex="-1"><a class="header-anchor" href="#_3-1-定义"><span>3.1 定义</span></a></h3><p>对于连续性随机变量，其概率分布函数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>可以表达成函数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>的积分</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msubsup space="4"><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.647em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-spacer style="margin-top:1.564em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c221E"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow data-mjx-texclass="ORD"><mi>x</mi></mrow></msubsup><msub><mi>f</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>d</mi><mi>u</mi></math></mjx-assistive-mml></mjx-container><p>称<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>为随机变量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></mjx-assistive-mml></mjx-container>的概率密度函数，可以理解为随机变量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></mjx-assistive-mml></mjx-container>出现在区间<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>+</mo><mi>d</mi><mi>x</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container>之间的概率为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></math></mjx-assistive-mml></mjx-container></p><h3 id="_3-2-特性" tabindex="-1"><a class="header-anchor" href="#_3-2-特性"><span>3.2 特性</span></a></h3><ul><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></mstyle><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2A7E TEX-A"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c2200"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2208"></mjx-c></mjx-mo><mjx-TeXAtom space="4" texclass="ORD"><mjx-mi class="mjx-ds mjx-b"><mjx-c class="mjx-c211D TEX-A"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⩾</mo><mn>0</mn><mo>,</mo><mi mathvariant="normal">∀</mi><mi>x</mi><mo>∈</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">R</mi></mrow></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mstyle><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D443 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2A7D TEX-A"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msubsup space="4"><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.647em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-spacer style="margin-top:1.665em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mi>P</mi><mo stretchy="false">(</mo><mi>a</mi><mo><</mo><mi>x</mi><mo>⩽</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mi>a</mi><mrow data-mjx-texclass="ORD"><mi>b</mi></mrow></msubsup><msub><mi>f</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>d</mi><mi>u</mi></mstyle></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mstyle><mjx-msubsup><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.647em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c221E"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-spacer style="margin-top:1.514em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c221E"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D462 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow data-mjx-texclass="ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><msub><mi>f</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>d</mi><mi>u</mi><mo>=</mo><mn>1</mn></mstyle></math></mjx-assistive-mml></mjx-container></li></ul><h3 id="_3-3-举例" tabindex="-1"><a class="header-anchor" href="#_3-3-举例"><span>3.3 举例</span></a></h3><p>对于均匀分布在[a,b]之间的随机变量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D448 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math></mjx-assistive-mml></mjx-container>，有</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D448 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c7B TEX-S3"></mjx-c></mjx-mo><mjx-mtable style="min-width:9.156em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-bottom:0.1em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-bottom:0.1em;"><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2A7D TEX-A"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2A7D TEX-A"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:left;padding-right:0.5em;padding-top:0.1em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0.5em;padding-top:0.1em;"><mjx-TeXAtom texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c74"></mjx-c><mjx-c class="mjx-c68"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c72"></mjx-c><mjx-c class="mjx-c77"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c65"></mjx-c></mjx-mtext></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n" style="vertical-align:0.25em;"></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>f</mi><mi>U</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">{</mo><mtable columnalign="left left" columnspacing="1em" rowspacing=".2em"><mtr><mtd><mn>1</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo></mtd><mtd><mrow data-mjx-texclass="ORD"><mi>a</mi><mo>⩽</mo><mi>x</mi><mo>⩽</mo><mi>b</mi></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mrow data-mjx-texclass="ORD"><mtext>otherwise</mtext></mrow></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo></mrow></math></mjx-assistive-mml></mjx-container><p>正态分布</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msqrt><mjx-sqrt><mjx-surd><mjx-mo class="mjx-n"><mjx-c class="mjx-c221A"></mjx-c></mjx-mo></mjx-surd><mjx-box style="padding-top:0.169em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi></mjx-box></mjx-sqrt></mjx-msqrt></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mfrac><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-msup size="s"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c221E"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c221E"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><msqrt><mn>2</mn><mi>π</mi></msqrt></mfrac></mstyle><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac></mrow></msup><mo>,</mo><mo>−</mo><mi mathvariant="normal">∞</mi><mo><</mo><mi>x</mi><mo><</mo><mo>+</mo><mi mathvariant="normal">∞</mi></math></mjx-assistive-mml></mjx-container><h2 id="_4-期望值" tabindex="-1"><a class="header-anchor" href="#_4-期望值"><span>4. 期望值</span></a></h2><h3 id="_4-1-定义" tabindex="-1"><a class="header-anchor" href="#_4-1-定义"><span>4.1 定义</span></a></h3><p>对于离散随机变量，定义期望值</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-lop" space="4"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>=</mo><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><msub><mi>x</mi><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub></mrow></math></mjx-assistive-mml></mjx-container><p>对于连续型随机变量，定义期望值</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msubsup space="4"><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.647em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c221E"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-spacer style="margin-top:1.514em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c221E"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>=</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow data-mjx-texclass="ORD"><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>x</mi><msub><mi>f</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></math></mjx-assistive-mml></mjx-container><p>比如对于扔色子的结果</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c36"></mjx-c></mjx-mn></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="3"><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c2E"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>=</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>5</mn><mo>+</mo><mn>6</mn><mo stretchy="false">)</mo><mo>=</mo><mn>3.5</mn></math></mjx-assistive-mml></mjx-container><p>对于标准正态分布</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></math></mjx-assistive-mml></mjx-container><h3 id="_4-2-特性" tabindex="-1"><a class="header-anchor" href="#_4-2-特性"><span>4.2 特性</span></a></h3><p>如果<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math></mjx-assistive-mml></mjx-container>是常量，那么</p><ul><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mi>X</mi><mo>+</mo><mi>c</mi><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>+</mo><mi>c</mi></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mi>c</mi><mi>X</mi><mo stretchy="false">]</mo><mo>=</mo><mi>c</mi><mi>E</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D44C TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44C TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo stretchy="false">]</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>+</mo><mi>E</mi><mo stretchy="false">[</mo><mi>Y</mi><mo stretchy="false">]</mo></math></mjx-assistive-mml></mjx-container></li></ul><h2 id="_5-大数定理" tabindex="-1"><a class="header-anchor" href="#_5-大数定理"><span>5. 大数定理</span></a></h2><p>对于随机变量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></mjx-assistive-mml></mjx-container>，获取其序列值<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.024em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.024em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2E"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2E"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2E"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.024em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><msub><mi>X</mi><mn>2</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><msub><mi>X</mi><mi>n</mi></msub></math></mjx-assistive-mml></mjx-container>，当<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mi class="mjx-n" space="4"><mjx-c class="mjx-c221E"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo stretchy="false">→</mo><mi mathvariant="normal">∞</mi></math></mjx-assistive-mml></mjx-container>时，其平均值趋近于期望值<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D443 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mstyle><mjx-munder><mjx-row><mjx-base style="padding-left:0.5em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6D"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.188em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c221E"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder><mjx-mstyle space="2"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.024em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.024em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22EF"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.024em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mstyle displaystyle="true" scriptlevel="0"><munder><mo data-mjx-texclass="OP" movablelimits="true">lim</mo><mrow data-mjx-texclass="ORD"><mi>n</mi><mo stretchy="false">→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munder><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>n</mi></mfrac></mstyle><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>+</mo><msub><mi>X</mi><mn>2</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mstyle></math></mjx-assistive-mml></mjx-container><h2 id="_6-蒙特卡洛积分法" tabindex="-1"><a class="header-anchor" href="#_6-蒙特卡洛积分法"><span>6. 蒙特卡洛积分法</span></a></h2><h3 id="_6-1-定义" tabindex="-1"><a class="header-anchor" href="#_6-1-定义"><span>6.1 定义</span></a></h3><p>想求一个函数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>在[a,b]区间的定积分</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43C TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-msubsup><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.647em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-spacer style="margin-top:1.665em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>I</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow><mrow data-mjx-texclass="ORD"><mi>b</mi></mrow></msubsup><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mstyle></math></mjx-assistive-mml></mjx-container><p>定义一个随机数序列<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.024em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msub><mi>X</mi><mi>i</mi></msub><mo fence="false" stretchy="false">}</mo></math></mjx-assistive-mml></mjx-container>，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.024em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>i</mi></msub></math></mjx-assistive-mml></mjx-container>在[a,b]区间的概率密度函数为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>X</mi></msub></math></mjx-assistive-mml></mjx-container>,定义</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi>g</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>f</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mstyle></math></mjx-assistive-mml></mjx-container><p>那么序列<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.024em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msup><mi>g</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></math></mjx-assistive-mml></mjx-container>也是一组随机数序列，其期望值</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.024em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msubsup space="4"><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.647em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-spacer style="margin-top:1.665em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-msup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msubsup space="4"><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.647em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-spacer style="margin-top:1.665em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D43C TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">[</mo><msup><mi>g</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow><mrow data-mjx-texclass="ORD"><mi>b</mi></mrow></msubsup><msup><mi>g</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mi>f</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow><mrow data-mjx-texclass="ORD"><mi>b</mi></mrow></msubsup><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mi>I</mi></math></mjx-assistive-mml></mjx-container><p>根据大数定理</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.024em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-munder space="4"><mjx-row><mjx-base style="padding-left:0.5em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c6C"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6D"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.188em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c221E"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder><mjx-mstyle space="2"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-munderover space="2"><mjx-over style="padding-bottom:0.192em;padding-left:0.51em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.148em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder></mjx-box></mjx-munderover><mjx-msup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mo class="mjx-var" size="s"><mjx-c class="mjx-c2032"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.024em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo stretchy="false">[</mo><msup><mi>g</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><munder><mo data-mjx-texclass="OP" movablelimits="true">lim</mo><mrow data-mjx-texclass="ORD"><mi>n</mi><mo stretchy="false">→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munder><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>n</mi></mfrac></mstyle><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></munderover><msup><mi>g</mi><mo data-mjx-alternate="1">′</mo></msup><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><p>所以</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mstyle><mjx-msubsup><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.647em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-spacer style="margin-top:1.665em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2248"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-munderover space="2"><mjx-over style="padding-bottom:0.2em;padding-left:0.408em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.148em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder></mjx-box></mjx-munderover><mjx-mstyle space="2"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D454 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.024em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.024em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mstyle displaystyle="true" scriptlevel="0"><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow><mrow data-mjx-texclass="ORD"><mi>b</mi></mrow></msubsup><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>≈</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>N</mi></mfrac></mstyle><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></munderover><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>g</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mi>f</mi><mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></mstyle></mstyle></math></mjx-assistive-mml></mjx-container><h3 id="_6-2举例" tabindex="-1"><a class="header-anchor" href="#_6-2举例"><span>6.2举例</span></a></h3><p>比如需要计算</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43C TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msubsup space="4"><mjx-mo class="mjx-lop"><mjx-c class="mjx-c222B TEX-S2"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;margin-left:-0.388em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.647em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom><mjx-spacer style="margin-top:1.513em;"></mjx-spacer><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-msup space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c38"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>I</mi><mo>=</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msubsup><msup><mi>x</mi><mn>2</mn></msup><mi>d</mi><mi>x</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>8</mn><mn>3</mn></mfrac></mstyle></math></mjx-assistive-mml></mjx-container><p>设随机变量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></mjx-assistive-mml></mjx-container>在[0,2]区间均匀分布，那么<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>X</mi></msub><mo>=</mo><mn>1</mn><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn></math></mjx-assistive-mml></mjx-container>,那么</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D43C TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2248"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-munderover space="2"><mjx-over style="padding-bottom:0.2em;padding-left:0.408em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D441 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.148em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder></mjx-box></mjx-munderover><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:-0.024em;"><mjx-mn class="mjx-n" size="s" style="margin-left:0.106em;"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-spacer style="margin-top:0.193em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>I</mi><mo>≈</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mi>N</mi></mfrac></mstyle><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></munderover><mn>2</mn><msubsup><mi>X</mi><mi>i</mi><mn>2</mn></msubsup></math></mjx-assistive-mml></mjx-container><p>用Mathmatica模拟</p><figure><img src="/assets/prob_01-D18Wt_Zs.gif" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><h2 id="_7-采样函数" tabindex="-1"><a class="header-anchor" href="#_7-采样函数"><span>7. 采样函数</span></a></h2><p>计算机中生成的随机数一般是概率均匀分布的随机变量，一般用<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D709 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ξ</mi></math></mjx-assistive-mml></mjx-container>表示均匀分布在[0,1]之间的随机变量，如果一个随机变量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></mjx-assistive-mml></mjx-container>的概率密度函数为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D453 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.06em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>X</mi></msub></math></mjx-assistive-mml></mjx-container>，概率分布函数为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>X</mi></msub></math></mjx-assistive-mml></mjx-container>，那么产生随机变量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></mjx-assistive-mml></mjx-container>的采样函数为概率分布函数的反函数</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.308em;margin-left:-0.106em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.228em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom><mjx-spacer style="margin-top:0.18em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D709 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msubsup><mi>F</mi><mi>X</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><h3 id="_7-1-证明" tabindex="-1"><a class="header-anchor" href="#_7-1-证明"><span>7.1 证明</span></a></h3><p>随机变量<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D709 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ξ</mi></math></mjx-assistive-mml></mjx-container>有一个很重要的特性</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D443 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D709 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2A7D TEX-A"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo>⩽</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi></math></mjx-assistive-mml></mjx-container><p>随机变量的概率的分布函数<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>X</mi></msub></math></mjx-assistive-mml></mjx-container>肯定是单调递增函数，所以它的反函数也一定是单调递增函数，所以</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:18.43em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D443 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.308em;margin-left:-0.106em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.228em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom><mjx-spacer style="margin-top:0.18em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D709 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2A7D TEX-A"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D443 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msubsup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.308em;margin-left:-0.106em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.228em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom><mjx-spacer style="margin-top:0.18em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D709 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2A7D TEX-A"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D443 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D709 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2A7D TEX-A"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D44B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D465 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>F</mi><mi>X</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo>⩽</mo><mi>x</mi><mo stretchy="false">)</mo></mtd><mtd><mi></mi><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><msubsup><mi>F</mi><mi>X</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⩽</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo>⩽</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><h3 id="_7-2-举例" tabindex="-1"><a class="header-anchor" href="#_7-2-举例"><span>7.2 举例</span></a></h3><p>比如想产生一个随机点，均匀分布在一个半径为R的圆内，设产生的随机点的极坐标为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>ρ</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，那么<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi></math></mjx-assistive-mml></mjx-container>的概率分布函数</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.106em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D443 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2A7D TEX-A"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mspace></mjx-mspace></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mi>ρ</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo>⩽</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>r</mi><mn>2</mn></msup><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><msup><mi>R</mi><mn>2</mn></msup><mspace linebreak="newline"></mspace></math></mjx-assistive-mml></mjx-container><p>所以产生随机点的采样函数为</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:9.305em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msubsup space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D439 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.247em;margin-left:-0.106em;"><mjx-TeXAtom size="s" texclass="ORD" style="margin-left:0.228em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom><mjx-spacer style="margin-top:0.289em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D709 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D445 TEX-I"></mjx-c></mjx-mi><mjx-msqrt><mjx-sqrt><mjx-surd><mjx-mo class="mjx-lop"><mjx-c class="mjx-c221A TEX-S2"></mjx-c></mjx-mo></mjx-surd><mjx-box style="padding-top:0.452em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D709 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D70C TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-box></mjx-sqrt></mjx-msqrt><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D70B TEX-I"></mjx-c></mjx-mi><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D709 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi>ρ</mi></mtd><mtd><mi></mi><mo>=</mo><msubsup><mi>F</mi><mi>ρ</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>ξ</mi><mi>ρ</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>R</mi><msqrt><msub><mi>ξ</mi><mi>ρ</mi></msub></msqrt></mtd></mtr><mtr><mtd><mi>θ</mi></mtd><mtd><mi></mi><mo>=</mo><mn>2</mn><mi>π</mi><msub><mi>ξ</mi><mi>θ</mi></msub></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><figure><img src="/assets/rnd_02-Dj4gCoIh.gif" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure></div><!--[--><!--]--></div><footer 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