概率论与数理统计
2021-09-28 16:29:29 249 举报AI智能生成
走过路过不要错过，最全面的概率论笔记总结！！！
作者其他创作
大纲/内容
数字特征<br>
数学期望<br>
定义
离散型随机变量期望<br><span class="equation-text" data-index="0" data-equation="P\{X=x_i\}=p_i,且级数\sum_i x_ip_i绝对收敛,E(X)=\sum_i x_ip_i" contenteditable="false"><span></span><span></span></span>
连续型随机变量期望<br><span class="equation-text" data-index="0" data-equation="概率密度f(x)，\int_{-\infty}^{+\infty}xf(x)dx绝对收敛,E(X)=\int_{-\infty}^{+\infty}xf(x)dx" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="\sum_i x_ip_i" contenteditable="false"><span></span><span></span></span> 收敛，记为 <span class="equation-text" data-index="1" data-equation="E(X)" contenteditable="false"><span></span><span></span></span> 或 <span class="equation-text" data-index="2" data-equation="EX" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="E(X^2)简记为EX^2" contenteditable="false"><span></span><span></span></span>
函数期望<br>
一维随机变量<br><span class="equation-text" data-index="0" data-equation="Y=g(X)" contenteditable="false"><span></span><span></span></span><br>
复合离散随机变量 <span class="equation-text" data-index="0" data-equation="Y = g(X)" contenteditable="false"><span></span><span></span></span><br><span class="equation-text" data-index="1" data-equation="E(Y)=E[g(X)]=\sum_{i=1}^\infty g(x_i)p_i" contenteditable="false"><span></span><span></span></span><br>
sp：<span class="equation-text" data-index="0" data-equation="X^2" contenteditable="false"><span></span><span></span></span><br><span class="equation-text" data-index="1" data-equation="E(X^2)=\sum x_i^2p_i" contenteditable="false"><span></span><span></span></span><br>
复合连续型随机变量 【f(x)为X的密度函数】<br><span class="equation-text" data-index="0" data-equation="E(Y)=E[g(X)]=\int_{-\infty}^{+\infty} g(x)f(x)dx" contenteditable="false"><span></span><span></span></span><br>
sp：<span class="equation-text" data-index="0" data-equation="X^2" contenteditable="false"><span></span><span></span></span>、|X|<br><span class="equation-text" data-index="1" data-equation="E(X^2)=\int_{-\infty}^{+\infty} x^2f(x)dx; \\ E(|X|)=\int_{-\infty}^{+\infty} |x|f(x)dx;" contenteditable="false"><span></span><span></span></span><br>
二维随机变量<br><span class="equation-text" data-index="0" data-equation="Z=g(X,Y)" contenteditable="false"><span></span><span></span></span><br>
离散<br><span class="equation-text" data-index="0" data-equation="E(Z)=E[g(X,Y)]=\sum \sum g(x_i,y_i)p_{ij}" contenteditable="false"><span></span><span></span></span><br>
sp: <span class="equation-text" data-index="0" data-equation="Z= XY" contenteditable="false"><span></span><span></span></span>; <span class="equation-text" data-index="1" data-equation="Z=g(x)" contenteditable="false"><span></span><span></span></span><br><span class="equation-text" data-index="2" data-equation="E[XY]=\sum \sum x_iy_ip_{ij};\\E[g(X)]=\sum \sum g(x_i)p_{ij}" contenteditable="false"><span></span><span></span></span><br>
连续<br><span class="equation-text" data-index="0" data-equation="E(Z)=E[g(X,Y)]=\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} g(x,y)f(x,y)dx" contenteditable="false"><span></span><span></span></span><br>
性质<br>
<br><span class="equation-text" data-index="0" data-equation="E(C)=C" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="E(CX)=CE(X)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="E(X+C)=E(X)+C" contenteditable="false"><span></span><span></span></span>
线性运算可拆性<br><span class="equation-text" data-index="0" data-equation="E(X\pm Y)=E(X)\pm E(Y)" contenteditable="false"><span></span><span></span></span>
一般情况<br><span class="equation-text" data-index="0" data-equation="E(a_1X_1+...+a_nX_n)=a_1E(X_1)+...+a_nE(X_n)" contenteditable="false"><span></span><span></span></span><br>
独立随机变量【充分条件】<br><span class="equation-text" data-index="0" data-equation="XY不相关\iff E(XY)=E(X)E(Y)" contenteditable="false"><span></span><span></span></span><br>
一般情况<br><span class="equation-text" data-index="0" data-equation="E(X_1X_2...X_n)=E(X_1)E(X_2)...E(X_n)" contenteditable="false"><span></span><span></span></span><br>
加权平均不等式<br><span class="equation-text" data-index="0" data-equation="X\geq a(或X\leq a),则E(X)\geq a (或X\geq a)" contenteditable="false"><span></span><span></span></span>
方差<br>
定义
定义【前提D(X)存在】<br><span class="equation-text" data-index="0" data-equation="D(X)=E\{[X-E(X)]\}^2【方差】; \\\sqrt{D(X)}【标准/均方差】" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="DX=\sum(x_i-EX)^2p_i" contenteditable="false"><span></span><span></span></span>
计算公式
<br><span class="equation-text" data-index="0" data-equation="D(X)=E(X^2)-[E(X)]^2" contenteditable="false"><span></span><span></span></span>
推理记忆 <br>1. <font color="#B71C1C">E(X) </font>是一个<font color="#B71C1C">数</font> <br>2. 利用<font color="#B71C1C">期望线性运算</font><br><span class="equation-text" data-index="0" data-equation="D(X)=E\{[X-E(X)]^2\} = \\ E(X^2-2XE(X)+E(X)^2) = \\ E(X^2)-2E(X)E(X)+E(X)^2=\\ E(X^2)-[E(X)]^2" contenteditable="false"><span></span><span></span></span><br>
随机变量平方的期望=原方差+原期望的平方<br><span class="equation-text" data-index="0" data-equation="E(X^2)=D(X)+[E(X)]^2" contenteditable="false"><span></span><span></span></span>
性质
常数方差为 0<br><span class="equation-text" data-index="0" data-equation="D(C)=0；D(C)=E(C^2)-E(C)^2=0\\D(X)=0 \iff P\{X=E(X)\}=1" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="D(X+c)=D(X)\\D(aX+b)=D(aX)=a^2D(X)" contenteditable="false"><span></span><span></span></span>
多维变量方程<br><span class="equation-text" data-index="0" data-equation="D(X\pm Y)=D(X)+D(Y)\pm 2Cov(X,Y)" contenteditable="false"><span></span><span></span></span>
协方差<br><span class="equation-text" data-index="0" data-equation="Cov(X,Y)=E(XY)-E(X)E(Y)" contenteditable="false"><span></span><span></span></span><br>
随机变量 X,Y 独立<br><span class="equation-text" data-index="0" data-equation="Cov(X,Y)=0,D(X\pm Y)=D(X)+D(Y)" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="g(t)=E[(X-t)^2]\geq D(X)=E[(X-EX)^2]" contenteditable="false"><span></span><span></span></span>
常见分布的期望/方差<br>
一维离散型随机变量<br>
0-1分布
公式<br><span class="equation-text" data-index="0" data-equation="E(X)=p" contenteditable="false"><span></span><span></span></span>
证明<br><span class="equation-text" data-index="0" data-equation="E(X)=0\cdot(1-p)+1\cdot p=p" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="D(X)=p(1-p)" contenteditable="false"><span></span><span></span></span>
证明<br><span class="equation-text" data-index="0" data-equation="E(X^2)=0^2\cdot (1-p)+1^2 \cdot p=p" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="D(X)=E(X^2)-[E(X)]^2=p-p^2=p(1-p)" contenteditable="false"><span></span><span></span></span>
二项分布<br>
<br><span class="equation-text" data-index="0" data-equation="E(X)=np" contenteditable="false"><span></span><span></span></span>
证明<br><span class="equation-text" data-index="0" data-equation="E(X)=\sum_{k=0}^n x\cdot C_n^k (1-p)^kp^{n-k}=np\sum_{k=1}^n C_n^{k} (1-p)^{k-1}p^{n-k}=np(1-p+p)^{n-1}=np" contenteditable="false"><span></span><span></span></span><br>
二项式定理<br><span class="equation-text" data-index="0" data-equation="\sum_{k=0}^n C_n^k x^ky^{n-k}=(x+y)^n" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="D(X)=np(1-p)" contenteditable="false"><span></span><span></span></span>
泊松分布<br>
期望公式<br><span class="equation-text" data-index="0" data-equation="E(X)=\lambda" contenteditable="false"><span></span><span></span></span>
证明<br><span class="equation-text" data-index="0" data-equation="E(X)=\sum_0^\infty kP\{X=k\}=\sum_0^\infty k\frac{\lambda^k}{k!}e^{-\lambda}=\lambda e^{-\lambda}\sum_{k=1}^\infty \frac{\lambda^{k-1}}{(k-1)!}【泰勒级数】=\lambda" contenteditable="false"><span></span><span></span></span><br>
级数公式<br><span class="equation-text" data-index="0" data-equation="e^\lambda=\sum_{k=1}^\infty \frac{\lambda^{k-1}}{(k-1)!}" contenteditable="false"><span></span><span></span></span><br>
方差公式<br><span class="equation-text" data-index="0" data-equation="D(X)=\lambda" contenteditable="false"><span></span><span></span></span><br>
E(X^2)<br><span class="equation-text" data-index="0" data-equation="\sum_0^\infty k^2P\{x=k\}=\sum_0^\infty k^2\frac{\lambda^k}{k!}e^{-\lambda}=e^{-\lambda}\sum_0^\infty[k(k-1)+k]\frac{\lambda^k}{k!}=\lambda^2+\lambda" contenteditable="false"><span></span><span></span></span><br>
D(X) = E(X^2) - [E(X)]^2<br><span class="equation-text" data-index="0" data-equation="D(X)=\lambda^2+\lambda-\lambda^2=\lambda" contenteditable="false"><span></span><span></span></span><br>
几何分布<br>
公式<br><span class="equation-text" data-index="0" data-equation="E(X)=\frac{1}{p}" contenteditable="false"><span></span><span></span></span>
证明<br><span class="equation-text" data-index="0" data-equation="E(X)=\sum_{k=1}^\infty k(1-p)^{k-1}p=p[1+...+n(1-p)^{n-1}]=p\cdot \frac{1}{p^2}=\frac{1}{p}" contenteditable="false"><span></span><span></span></span>
级数公式<br><span class="equation-text" data-index="0" data-equation="1+...+nx^{n-1}+...=(x+...+x^n+...)'=(\frac{x}{1-x})'=\frac{1}{(1-x)^2}" contenteditable="false"><span></span><span></span></span>
1/(1-x) (-1<x<1) 【高等数学幂级数公式来源】<br><span class="equation-text" data-index="0" data-equation="1/(1-x)=1+x+x^2+...+x^n+...=\sum_{n=0}^\infty x^n" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="D(X)=\frac{1-p}{p^2}" contenteditable="false"><span></span><span></span></span>
E(X^2) 【中间证明又涉及级数，此次省略】<br><span class="equation-text" data-index="0" data-equation="E(X^2)=\sum_0^\infty k^2(1-p)^{k-1}p=\sum_0^\infty[(k+1)k(1-p)^{k-1}p]-E(X)=p\cdot\frac{2}{p^3}-\frac{1}{p}=\frac{2-p}{p^2}" contenteditable="false"><span></span><span></span></span><br>
一维连续型随机变量<br>
均匀分布<br><span class="equation-text" data-index="0" data-equation="X\sim U(a,b)" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="E(X)=\frac{a+b}{2}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)=\frac{1}{b-a},EX=\int_{-\infty}^{+\infty} xf(x)dx= \frac{x^2}{2(b-a)}|_a^b"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="D(X)=\frac{(b-a)^2}{12}" contenteditable="false"><span></span><span></span></span>
E(X^2)<br><span class="equation-text" data-index="0" data-equation="E(X^2)=(a^2+ab+b^2)/3" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="EX^2=\int_{-\infty}^{+\infty}x^2f(x)dx= \frac{x^3}{3(b-a)}|_a^b"><span></span><span></span></span>
指数分布<br><span class="equation-text" data-index="0" data-equation="X \sim E(\lambda)" contenteditable="false"><span></span><span></span></span><br>
期望公式<br><span class="equation-text" data-index="0" data-equation="E(X)=\frac{1}{\lambda}" contenteditable="false"><span></span><span></span></span><br>
证明<br><span class="equation-text" data-index="0" data-equation="E(X)=\int_{-\infty}^{+\infty}xf(x)dx=\int_0^{+\infty}x\lambda e^{-\lambda x}dx=\frac{1}{\lambda}\int_0^{+\infty}\lambda xe^{-\lambda x}d\lambda x（分部积分）=\frac{1}{\lambda}" contenteditable="false"><span></span><span></span></span>
方差公式<br><span class="equation-text" data-index="0" data-equation="D(X)=\frac{1}{\lambda^2}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="E(X^2)=\int_{-\infty}^{+\infty}x^2f(x)dx=\int_0^{+\infty}x\lambda e^{-\lambda x}dx=\frac{1}{\lambda^2}\int_0^{+\infty}(\lambda x)^2e^{-\lambda x}d\lambda x（分部积分）=\frac{2}{\lambda^2}" contenteditable="false"><span></span><span></span></span>
正态分布<br><span class="equation-text" data-index="0" data-equation="X \sim N(\mu,\sigma^2)" contenteditable="false"><span></span><span></span></span><br>
期望公式 <br><span class="equation-text" data-index="0" data-equation="E(X)=\mu" contenteditable="false"><span></span><span></span></span><br>
证明<br><span class="equation-text" data-index="0" data-equation="E(X)=\int_{-\infty}^{+\infty}x \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{\sigma^2}}【令t=\frac{x-\mu}{\sigma}】=\int_{-\infty}^{+\infty} \frac{\sigma t+\mu}{\sqrt{2\pi}\sigma}e^{-t^2}d(\sigma t+\mu)\\ ----\\=\sigma\int_{-\infty}^{+\infty} \frac{t}{\sqrt{2\pi}}e^{-t}dt（奇函数）+\mu\int_{-\infty}^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-t^2}dt（标准正态分布）=0+\mu=\mu" contenteditable="false"><span></span><span></span></span>
方差公式<br><span class="equation-text" data-index="0" data-equation="D(X)=\sigma^2" contenteditable="false"><span></span><span></span></span>
证明<br><span class="equation-text" data-index="0" data-equation="E(X^2)=\mu^2+\sigma^2; D(X)=E(X^2)-[E(X)]^2=\sigma^2" contenteditable="false"><span></span><span></span></span><br>
性质<br>
<br><span class="equation-text" data-index="0" data-equation="X\sim N(\mu,\sigma^2),则Y=aX+b\sim N(a\mu+b,a^2\sigma^2)【a\neq 0】" contenteditable="false"><span></span><span></span></span>
标准正态分布随机变量关系<br><span class="equation-text" data-index="0" data-equation="a=1/\sigma,b=-\mu/\sigma,则Y=\frac{X-\mu}{\sigma}\sim N(0,1)" contenteditable="false"><span></span><span></span></span>
独立正态的线性组合仍时正态分布【μ和σ^2同时线性组合】<br><span class="equation-text" data-index="0" data-equation="X\sim N(\mu,\sigma^2),X_1,...X_n相互独立，则a_i不全为0时,\sum_{i=1}^n a_iX_i\sim N(\sum a_i\mu_i,\sum a_i^2\sigma_i^2)" contenteditable="false"><span></span><span></span></span>
二维正态分布<br>
<br><span class="equation-text" data-index="0" data-equation="X\sim N(\mu_1,\sigma_1^2),Y\sim N(\mu_2,\sigma_2^2)" contenteditable="false"><span></span><span></span></span>
定义<br><span class="equation-text" data-index="0" data-equation="(X,Y)\sim N(\mu_1,\mu_2;\sigma_1^2,\sigma_2^2;\rho)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="aX+bY(a^2+b^2\neq 0)服从正态分布,sp: X \pm Y" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\begin{cases}U=a_1X+b_1Y\\V=a_2X+b_2Y\end{cases},当\begin{vmatrix}a_1&b_1\\a_2&b_2\end{vmatrix} \neq 0时，(U,V)服从二维正态分布" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="X与Y相互独立\iff \rho_{XY}=\rho=0 \iff X与Y不相关" contenteditable="false"><span></span><span></span></span>
随机变量函数<br><span class="equation-text" data-index="0" data-equation="Z = aX + bY" contenteditable="false"><span></span><span></span></span><br>
求解期望<br><span class="equation-text" data-index="0" data-equation="E(Z)=aE(X)+bE(Y)" contenteditable="false"><span></span><span></span></span>
求解方差<br><span class="equation-text" data-index="0" data-equation="D(X\pm Y)=a^2D(X)+b^2D(Y)\pm 2Cov(X,Y)" contenteditable="false"><span></span><span></span></span>
求解协方差<br><span class="equation-text" data-index="0" data-equation="Cov(X,Y)=\rho_{XY}\sqrt{D(X)}\sqrt{D(Y)}" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="X_1,X_2相互独立且服从N(\mu,\sigma^2)，则D(X_1X_2)="><span></span><span></span></span>
协方差
定义
<br><span class="equation-text" data-index="0" data-equation="Cov(X,Y)=E\{[X-E(X)][Y-E(Y)]\}" contenteditable="false"><span></span><span></span></span>
利用期望求解<br><span class="equation-text" data-index="0" data-equation="Cov(X,Y)=E(XY)-E(X)E(Y)" contenteditable="false"><span></span><span></span></span>
证明<br><span class="equation-text" data-index="0" data-equation="E{XY-YE(X)-XE(Y)+E(X)E(Y)}=\\E(XY)-E(X)E(Y)-E(X)E(Y)+E(X)E(Y)=\\E(XY)-E(X)E(Y)" contenteditable="false"><span></span><span></span></span><br>
利用 相关系数/方差 求解<br><span class="equation-text" data-index="0" data-equation="Cov(X,Y)=\sqrt{D(X)}\sqrt{D(Y)}\rho_{XY}【D(X),D(Y)>0】" contenteditable="false"><span></span><span></span></span>
性质<br>
<br><span class="equation-text" data-index="0" data-equation="Cov(X,X)=D(X)=E(X^2)-[E(X)]^2" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="Cov(Y,X)=Cov(X,Y)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="Cov(X,C)=0【C为常数】;C-E(C)=0" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="Cov(aX,bY)=abCov(X,Y)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="Cov(X_1\pm X_2,Y)=Cov(X_1,Y)\pm Cov(X_2,Y)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="Cov(\sum a_iX_i,\sum b_jY_j)=\sum_{i=1}^m \sum_{j=1}^n a_ib_j Cov(X_i,Y_j)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="D(X\pm Y)=D(X)+D(Y)\pm 2Cov(X,Y)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="D(X+ Y+Z)=D(X)+D(Y)+D(Z)+ 2Cov(X,Y)+2Cov(Y,Z)+2Cov(X,Z)" contenteditable="false"><span></span><span></span></span>
相关系数<br>
定义
<br><span class="equation-text" data-index="0" data-equation="\rho_{XY}=\frac{Cov(X,Y)}{\sqrt{D(X)}\sqrt{D(Y)}}【D(X),D(Y)>0】" contenteditable="false"><span></span><span></span></span>
性质<br>
范围绝对值为1<br><span class="equation-text" data-index="0" data-equation="\rho_{XY}\in[-1,1]" contenteditable="false"><span></span><span></span></span>
X 与 Y 不相关 ≠ 独立<br>
<br><span class="equation-text" data-index="0" data-equation="\rho_{XY}=0\iff Cov(X,Y)=0\iff D(X\pm Y)=D(X)+D(Y)\iff E(XY)=E(X)E(Y)" contenteditable="false"><span></span><span></span></span>
相关系数 |ρ| = 1 充要条件<br><span class="equation-text" data-index="0" data-equation="|\rho_{XY}|=1\iff P\{Y=aX+b\}=1(a\neq 0),\rho_{XY}=\begin{cases}1,a>0\\-1,a<0\ \end{cases}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="X 和 Y 独立\implies有 X 和 Y不相关【充分非必要】" contenteditable="false"><span></span><span></span></span>
(X,Y）服从正态分布，此时等价<br>
矩<br>
k阶原点矩<br>
<br><span class="equation-text" data-index="0" data-equation="E(X^k),k=1时为期望" contenteditable="false"><span></span><span></span></span>
k阶中心矩<br>
<br><span class="equation-text" data-index="0" data-equation="E\{[X-E(X)]^k\},k=2时为方差,k-1时，E\{[X-E(X)]\}=0" contenteditable="false"><span></span><span></span></span>
k+l 阶混合矩<br>
<br><span class="equation-text" data-index="0" data-equation="E(X^kY^l)=E\{[X-E(X)]^k[Y-E(Y)]^l\}" contenteditable="false"><span></span><span></span></span>
随机变量标准化<br>
<br><span class="equation-text" data-index="0" data-equation="X^*=\frac{X-E(X)}{\sqrt{D(X)}},D(X)\neq 0,E(X^*)=0,D(X^*)=1" contenteditable="false"><span></span><span></span></span>
数字特征与独立性
独立，则协方差 = 0
大数定理和中心极限定理<br>
依概率收敛与大数定律<br>
依概率收敛<br>
定义<br><span class="equation-text" data-index="0" data-equation="\lim_{n\rightarrow \infty}P\{|Y_n-a|<\xi\}=1 或\lim_{n\rightarrow \infty}\{|Y_n-a| \geq \xi \}=0" contenteditable="false"><span></span><span></span></span>
记为<br><span class="equation-text" data-index="0" data-equation="Y_n \underrightarrow{\space P\space} a." contenteditable="false"><span></span><span></span></span><br>
大数定理【条件不背】<br>
切比雪夫大数定理<br>
<span class="equation-text" data-index="0" data-equation="X_1,..,X_n" contenteditable="false"><span></span><span></span></span>两两不相关随机变量序列，<span class="equation-text" data-index="1" data-equation="EX_i,DX_i" contenteditable="false"><span></span><span></span></span>均存在，存在常数C<span class="equation-text" data-index="2" data-equation="，D(X_i)\leq C" contenteditable="false"><span></span><span></span></span>, <br><span class="equation-text" data-index="3" data-equation=" 则\forall \xi>0 \lim_{n\rightarrow \infty} P\{| \frac{1}{n} \sum X_i-\frac{1}{n} EX_i |\leq \xi\}=1" contenteditable="false"><span></span><span></span></span><br>
伯努利大数定理
随机变量 <span class="equation-text" data-index="0" data-equation="X_n \sim B(n,p) , 则 \forall \xi > 0, \lim_{n \rightarrow \infty} P\{| \frac{X_n}{n}-p | \leq \xi\}=1" contenteditable="false"><span></span><span></span></span>
辛钦大数定理<br>
<span class="equation-text" data-index="0" data-equation="X_1,..,X_n" contenteditable="false"><span></span><span></span></span> 独立同分布 <span class="equation-text" data-index="1" data-equation="EX_i = \mu, \forall \xi > 0 \lim_{n \rightarrow \infty} P\{| \sum X_i-\mu | \leq \xi\}=1" contenteditable="false"><span></span><span></span></span>
中心极限定理
独立同分布<br>列维 —— 林格伯格定理<br>
<span class="equation-text" data-index="0" data-equation="\lim_{n \rightarrow \infty} P\{| \frac{X_n-n\mu}{\sqrt{n}\sigma }\leq x | \}=\Phi(x)" contenteditable="false"><span></span><span></span></span>
正态分布为极限分布<br>棣莫弗 —— 拉普拉斯极限中心定理<br>
<span class="equation-text" data-index="0" data-equation="\lim_{n \rightarrow \infty} P\{| \frac{X_n-np}{\sqrt{np(1-p)}}\leq x | \leq \xi\}=\Phi(x)" contenteditable="false"><span></span><span></span></span>
切比雪夫不等式<br>
公式<br><span class="equation-text" data-index="0" data-equation="P\{|X-\mu|\geq \xi\}\leq \frac{\sigma^2}{\xi^2}或P\{|X-\mu|< \xi\}\geq1- \frac{\sigma^2}{\xi^2}【E(X)=\mu,D(X)=\sigma^2】" contenteditable="false"><span></span><span></span></span><br>
形象记忆<br><span class="equation-text" data-index="0" data-equation="\geq \xi\leq【么么哒】" contenteditable="false"><span></span><span></span></span><br>
数理统计的基本概念<br>
总体
样本
定义<br>
样本值 / 观察值<br><span class="equation-text" data-index="0" data-equation="(x_1,x_2,...,x_n)" contenteditable="false"><span></span><span></span></span>为观察值【抽样后】<br>
样本 / 简单随机变量<br><span class="equation-text" data-index="0" data-equation="(X_1,X_2,...,X_n)" contenteditable="false"><span></span><span></span></span>为n维随机变量【抽样前】<br>
前提条件<br><span class="equation-text" data-index="0" data-equation="X_1,...,X_n " contenteditable="false"><span></span><span></span></span>相互独立且与总体<span class="equation-text" data-index="1" data-equation="X" contenteditable="false"><span></span><span></span></span>同分布<br>
样本数字特征性质<br>
Xi 与 X同分布<br>
<span class="equation-text" data-index="0" data-equation="E(X_i)=E(X)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="D(X_i)=D(X)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="F_{X_i}(x)=F_X(x)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="E(\bar X)=\mu" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="D(\bar X)=\frac{D(X)}{N}=\frac{\sigma^2}{n}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="E(S^2)=E[\frac{1}{n-1}\sum_{i=1}^n(X_i-\bar X)^2]=DX=\sigma^2" contenteditable="false"><span></span><span></span></span>
统计量
定义<br>
不含任何未知数<br>样本不含任何参数的函数 <span class="equation-text" data-index="0" data-equation="g(X_1,...,X_n)" contenteditable="false"><span></span><span></span></span><br>
数字特征
样本均值
<br><span class="equation-text" data-index="0" data-equation="\overline X=\frac{1}{n}\sum X_i" contenteditable="false"><span></span><span></span></span>
样本方差
<br><span class="equation-text" data-index="0" data-equation="S^2=\frac{1}{n-1}\sum_i^n (X_i-\bar X)^2=\frac{1}{n-1}(\sum_i^n X_i^2-n\bar X^2)" contenteditable="false"><span></span><span></span></span>
样本标准差
<br><span class="equation-text" data-index="0" data-equation="S=\sqrt{S^2}=\sqrt{\frac{1}{n-1}(\sum_i^n X_i^2-n\bar X^2)}" contenteditable="false"><span></span><span></span></span>
样本 k 阶原点矩<br>
<br><span class="equation-text" data-index="0" data-equation="A_k=\frac{1}{n}\sum_i^n X_i^k,A_1=\bar X" contenteditable="false"><span></span><span></span></span>
样本 k 阶中心矩<br>
<br><span class="equation-text" data-index="0" data-equation="B_k=\frac{1}{n}\sum_i^n(X_i-\bar X)^k,B_1=0,B_2=\frac{n-1}{n}S^2" contenteditable="false"><span></span><span></span></span>
顺序统计量<br>
<br><span class="equation-text" data-index="0" data-equation="X_1^*=min\{X_1,...,X_n\};X_n^*=max\{X_1,...,X_n\}" contenteditable="false"><span></span><span></span></span>
结论<br>
正态分布方差<br><span class="equation-text" data-index="0" data-equation="E(\bar X)=\mu,D(\bar X)=\frac{\sigma^2}{n},E(S^2)=\sigma" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="E\sum_{i=1}^n(X_i-\bar X)^2=E(n-1)S^2=(n-1)\sigma^2" contenteditable="false"><span></span><span></span></span>
上侧分位点<br>
定义<br><span class="equation-text" data-index="0" data-equation="设随机变量U\sim N(0,1)，对于给定正整数a,0<a<1，\\则称满足P\{U\geq U_a\}=a的点U_a(点的值为U_a)为上侧分位点" contenteditable="false"><span></span><span></span></span><br>
理解<br><span class="equation-text" data-index="0" data-equation="只需要明白随机变量X在大于x的区间部分概率为a,这个点记作U_a 即可" contenteditable="false"><span></span><span></span></span><br>
分布<br>
卡方分布<br><span class="equation-text" data-index="0" data-equation="\chi^2分布" contenteditable="false"><span></span><span></span></span>
定义<br>
条件【<font color="#B71C1C">独立变量</font>必须服从<font color="#B71C1C">标准正态</font>分布】<br><span class="equation-text" data-index="0" data-equation="n个独立X\sim N(0,1)的平方和" contenteditable="false"><span></span><span></span></span><br>
自由度为n的卡方分布<br><span class="equation-text" data-index="0" data-equation="\chi^2=X_1^2+...+X_n^2,记作\chi^2\sim \chi^2(n)" contenteditable="false"><span></span><span></span></span>
性质
期望、方差
<br><span class="equation-text" data-index="0" data-equation="\chi^2\sim \chi^2(n):E(X^2)=n,D(X^2)=2n" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\chi^2\sim \chi^2(1):E(X^2)=1,D(X^2)=2" contenteditable="false"><span></span><span></span></span>
可加性<br>
<br><span class="equation-text" data-index="0" data-equation="\chi^2\sim \chi^2(n),且\chi_1^2和\chi_2^2相互独立则，\chi_1^2 + \chi_2^2\sim \chi^2(n_1+n_2)" contenteditable="false"><span></span><span></span></span>
证明<br>
方差<br><span class="equation-text" data-index="0" data-equation="E(X)=\mu=0,D(X)=\sigma^2=1,E(X^2)=[E(X)]^2+D(X)=1\\E(X^4)=...=0+3E(X^2)=3,D(X^2)=3-1^2=2" contenteditable="false"><span></span><span></span></span><br>
上侧 α 分位点
<br><span class="equation-text" data-index="0" data-equation="P\{\chi^2>\chi_\alpha^2 (n)\}=\alpha" contenteditable="false"><span></span><span></span></span>
t 分布【student分布】<br><span class="equation-text" data-index="0" data-equation="T\sim t(n)" contenteditable="false"><span></span><span></span></span><br>
条件<br>
<br><span class="equation-text" data-index="0" data-equation="X \sim N(0,1),Y\sim \chi^2(n)，且X和Y相互独立" contenteditable="false"><span></span><span></span></span>
定义<br>
<br><span class="equation-text" data-index="0" data-equation="T=\frac{X}{\sqrt{Y/n}}\sim t(n)" contenteditable="false"><span></span><span></span></span>
性质<br>
分布具有对称性<br>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="t^2(n)=F(1,n)"><span></span><span></span></span>
上侧 α 分位点
F 分布<br><span class="equation-text" data-index="0" data-equation="F\sim F(n_1,n_2)" contenteditable="false"><span></span><span></span></span><br>
定义<br>
n_1第一自由度，n_2第二自由度的 F分布<br><span class="equation-text" data-index="0" data-equation="X\sim \chi^2(n_1),Y\sim \chi^2(n_2),且X和Y相互独立，则称F=\frac{X/n_1}{Y/n_2} " contenteditable="false"><span></span><span></span></span>
性质
<br><span class="equation-text" data-index="0" data-equation="F \sim F(n_1,n_2),则 \frac{1}{F}=\frac{Y/n_2}{X/n_1} \sim F(n_2,n_1)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="T\sim t(n),则T^2\sim F(1,n);\frac{1}{T^2}\sim F(n,1)" contenteditable="false"><span></span><span></span></span>
当 n1,n2 为 1 时<br><span class="equation-text" data-index="0" data-equation="F=X/Y" contenteditable="false"><span></span><span></span></span><br>
上侧 α 分位点
<br><span class="equation-text" data-index="0" data-equation="F_{1-\alpha}(n_1,n_2)=\frac{1}{F_\alpha(n_2,n_1)}" contenteditable="false"><span></span><span></span></span>
<font color="#B71C1C">单正态总体</font><font color="#B71C1C">的抽样分布</font><br>
前提<br>
<span class="equation-text" data-index="0" data-equation="设 X_1,..X_n为正太总体N(\mu,\sigma^2)的简单随机样本，\bar X为样本均值，S^2为样本方差" contenteditable="false"><span></span><span></span></span>
结论
<br><span class="equation-text" data-index="0" data-equation="\overline X=\frac{1}{n}(X_1+X_2+..+X_n)\sim N(\mu,\frac{\sigma^2}{n})或U=\frac{\bar X-\mu}{\sigma/\sqrt{n}}\sim N(0,1)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\overline X与 S^2 相互独立，且\chi^2=\frac{(n-1)S^2}{\sigma^2}=\frac{\sum_{i=1}^n (X_i-\bar X)^2}{\sigma^2}\sim \chi^2(n-1)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\chi^2=\frac{\sum_{i=1}^n (X_i-\mu)^2}{\sigma^2}\sim \chi^2(n)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="T=\frac{\bar X-\mu}{S/\sqrt{n}}\sim t(n-1)" contenteditable="false"><span></span><span></span></span>
证明<br><span class="equation-text" data-index="0" data-equation="T=\frac{\bar X-\mu}{S/\sqrt{n}}=\frac{\frac{\bar X-\mu}{\sigma/\sqrt{n}}}{\sqrt{\frac{(n-1)S^2}{\sigma^2}/(n-1)}}\sim t(n-1)" contenteditable="false"><span></span><span></span></span><br>
结论2<br>
<br><span class="equation-text" data-index="0" data-equation="n\bar X \sim N(n\mu,n\sigma^2)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{(n-1)S^2}{\sigma^2}\sim \chi^2(n-1)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\frac{(\bar X-\mu)}{S/\sqrt{n}} \sim t(n-1)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="t(n-1)^2\sim F(1,n-1)" contenteditable="false"><span></span><span></span></span>
双正态总体的抽样分布<br>
前提<br><span class="equation-text" data-index="0" data-equation="(X1,...,Xn),(Y1,...,Yn)相互独立" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="T=\frac{(\bar X-\bar Y)-(\mu_1-\mu_2)}{S_w\sqrt{1/n_1+1/n_2}}\sim t(n_1+n_2-2); S_w=\sqrt{\frac{(n_1-1)S_1^2+(n_2-1)S_2^2}{n_1+n_2-2}}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="F=\frac{S_1^2/\sigma_1^2}{S_2^2/\sigma_2^2}\sim F(n_1-1,n_2-1)" contenteditable="false"><span></span><span></span></span>
参数估计<br>
基本概念
点估计
定义
点估计定义<br>构造一个统计量 <span class="equation-text" data-index="0" data-equation="\hat \theta(X_1,...,X_n)," contenteditable="false"><span></span><span></span></span> 用来估计总体X分布中含有的未知数<br>
总体未知参数<br><span class="equation-text" data-index="0" data-equation="\theta=(\theta_1,...,\theta_k)" contenteditable="false"><span></span><span></span></span><br>
统计量<br><span class="equation-text" data-index="0" data-equation="\hat{\theta}(X_1,...,X_n)" contenteditable="false"><span></span><span></span></span><br>
无偏性
无偏估计
定义<br>设<span class="equation-text" data-index="0" data-equation="\hat \theta"><span></span><span></span></span>为<span class="equation-text" data-index="1" data-equation="\theta"><span></span><span></span></span>的估计量,若<span class="equation-text" data-index="2" data-equation="E(\hat \theta)=\theta"><span></span><span></span></span>,则称<span class="equation-text" data-index="3" data-equation="\hat \theta为\theta"><span></span><span></span></span> 的无偏估计.<span class="equation-text" data-index="4" data-equation="\lim_{n\rightarrow \infty}E(\hat \theta)=\theta"><span></span><span></span></span>,称为渐近
结论<br>
样本与总体<br><span class="equation-text" data-index="0" data-equation="\bar X为E(X)=\mu" contenteditable="false"><span></span><span></span></span> 的无偏估计<span class="equation-text" data-index="1" data-equation=",E(\bar X)=E(X)=\mu" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="S^2为D(X)=\sigma^2" contenteditable="false"><span></span><span></span></span> 的无偏估计，<span class="equation-text" data-index="1" data-equation="E(S^2)=D(X)=\sigma^2" contenteditable="false"><span></span><span></span></span>
估计量<span class="equation-text" data-index="0" data-equation="\hat \theta_i" contenteditable="false"><span></span><span></span></span>均为<span class="equation-text" data-index="1" data-equation="\theta" contenteditable="false"><span></span><span></span></span> 的无偏估计，则其线性组合任为无偏估计
更有效估计
<span class="equation-text" data-index="0" data-equation="\theta " contenteditable="false"><span></span><span></span></span> 有两个无偏估计量 <span class="equation-text" data-index="1" data-equation="\hat \theta_1" contenteditable="false"><span></span><span></span></span> 和 <span class="equation-text" data-index="2" data-equation="\hat \theta_2" contenteditable="false"><span></span><span></span></span>， 当 <span class="equation-text" data-index="3" data-equation="D(\hat \theta_1)<D(\hat \theta_2), \hat \theta_1" contenteditable="false"><span></span><span></span></span> 比 <span class="equation-text" data-index="4" data-equation="\hat \theta_2" contenteditable="false"><span></span><span></span></span> 更有效<br>
一致估计<br>
<span class="equation-text" data-index="0" data-equation="\hat \theta(X_1,...,X_n)," contenteditable="false"><span></span><span></span></span> 其中 <span class="equation-text" data-index="1" data-equation="\hat \theta" contenteditable="false"><span></span><span></span></span> 依概率收敛于 <span class="equation-text" data-index="2" data-equation="\theta" contenteditable="false"><span></span><span></span></span> 则称其为 <span class="equation-text" data-index="3" data-equation="\theta" contenteditable="false"><span></span><span></span></span> 的一致估计量<br>
估计量/值
估计量<br>随机变量<br>
估计值<br>所取的具体值<br>
点估计<br>估计量的值估计未知参数<br>
估计方法
矩估计
总体矩<br>
一阶原点矩=期望 ，二中心阶矩=方差<br><span class="equation-text" data-index="0" data-equation="\mu_1=E(X),\gamma_2=D(X)" contenteditable="false"><span></span><span></span></span>
样本矩<br>
均值，和样本方差<br><span class="equation-text" data-index="0" data-equation="A_1=\bar X,B_2=\frac{1}{n}(\sum_{i=1}^n X^2_i-n\bar X^2)" contenteditable="false"><span></span><span></span></span>
关系 / 定义<br>
基于 <b>大数定律</b> 利用 <b>样本矩</b> 估计 <b>总体矩</b>
1) 对于总体 <span class="equation-text" data-index="0" data-equation="X " contenteditable="false"><span></span><span></span></span>的分布含有<span class="equation-text" data-index="1" data-equation="\theta_1,..,\theta_n" contenteditable="false"><span></span><span></span></span>, 有 <span class="equation-text" data-index="2" data-equation="\alpha_k=E(X^k)" contenteditable="false"><span></span><span></span></span> 存在，则<span class="equation-text" data-index="3" data-equation=" \alpha_k " contenteditable="false"><span></span><span></span></span>显然为关于 <span class="equation-text" data-index="4" data-equation="\theta_i" contenteditable="false"><span></span><span></span></span> 的函数<br>
2) 则样本k阶原点矩为 <span class="equation-text" data-index="0" data-equation="A_k = \frac{1}{n} \sum X_i^k" contenteditable="false"><span></span><span></span></span>
矩估计量表示
<br><span class="equation-text" data-index="0" data-equation="E(X)=g(\theta)=\bar X \implies \hat \theta=g^{-1}(\bar X)【反函数】" contenteditable="false"><span></span><span></span></span>
最大(极大)似然估计
似然函数<br>
样本取得观察值<span class="equation-text" data-index="0" data-equation="x_1,...,x_n" contenteditable="false"><span></span><span></span></span>的概率<span class="equation-text" data-index="1" data-equation=",L(\theta)" contenteditable="false"><span></span><span></span></span>
离散型<br>
<br><span class="equation-text" data-index="0" data-equation="P\{X=x\}=p(x;\theta),则L(\theta)=P\{X_1=x_1,...,X_n=x_n\}=\prod _{i=1}^nP\{X_i=x_i\}" contenteditable="false"><span></span><span></span></span>
连续型<br>
<br><span class="equation-text" data-index="0" data-equation="f(x)=f(x;\theta),则L(\theta)=P\{X_1\in U(x_1),...,X_n\in U(x_n)\}=\prod_{i=1}^n f(x_1;\theta)dx_i" contenteditable="false"><span></span><span></span></span>
求解<br>
思路】<br>在未知参数<span class="equation-text" data-index="0" data-equation="\theta" contenteditable="false"><span></span><span></span></span> 取值范围求<span class="equation-text" data-index="1" data-equation="\hat \theta,使L(\hat \theta)=maxL(\theta)" contenteditable="false"><span></span><span></span></span>，<br>简言之就是求<span class="equation-text" data-index="2" data-equation="L(\theta)" contenteditable="false"><span></span><span></span></span>函数的最大值点 <span class="equation-text" data-index="3" data-equation="\hat \theta" contenteditable="false"><span></span><span></span></span>, 使得<span class="equation-text" data-index="4" data-equation="D(\hat \theta)" contenteditable="false"><span></span><span></span></span>值最小，成为最有效估计量<br>
步骤<br>
写出对应函数<span class="equation-text" data-index="0" data-equation="L(\theta)" contenteditable="false"><span></span><span></span></span>,取对数<span class="equation-text" data-index="1" data-equation="\ln L(\theta)" contenteditable="false"><span></span><span></span></span>【对数求导法】
<span class="equation-text" data-index="0" data-equation="\frac{d L(\theta)}{d\theta}=0或\frac{d\ln L(\theta)}{d\theta}=0" contenteditable="false"><span></span><span></span></span>求出唯一驻点，<span class="equation-text" data-index="1" data-equation="\hat \theta=(\hat \theta_1,\hat \theta_2)" contenteditable="false"><span></span><span></span></span>
若方差无解，则取端点或边界，需要根据具体情况分析<br>
区间估计<br>
置信区间<br>
定义
设 <span class="equation-text" data-index="0" data-equation="\theta" contenteditable="false"><span></span><span></span></span> 是总体 <span class="equation-text" data-index="1" data-equation="X" contenteditable="false"><span></span><span></span></span> 的未知参数，<span class="equation-text" data-index="2" data-equation="X_1,..,X_n" contenteditable="false"><span></span><span></span></span> 是来自总体 <span class="equation-text" data-index="3" data-equation="X" contenteditable="false"><span></span><span></span></span> 的样本，<br>对于给定的<span class="equation-text" data-index="4" data-equation="\alpha (0<\alpha<1)" contenteditable="false"><span></span><span></span></span> , 如果两个统计量满足 <span class="equation-text" data-index="5" data-equation="P\{\theta_1<\theta<\theta_2 \}=1-\alpha" contenteditable="false"><span></span><span></span></span><br>则称随机区间<span class="equation-text" data-index="6" data-equation="(\theta_1,\theta_2)" contenteditable="false"><span></span><span></span></span>为<span class="equation-text" data-index="7" data-equation="\theta" contenteditable="false"><span></span><span></span></span>的<font color="#B71C1C">置信水平(置信度)</font>为 <span class="equation-text" data-index="8" data-equation="1-\alpha" contenteditable="false"><span></span><span></span></span> 的<font color="#B71C1C">置信区间(区间估计)</font>，<br>简称为 <span class="equation-text" data-index="9" data-equation="\theta" contenteditable="false"><span></span><span></span></span> 的 <span class="equation-text" data-index="10" data-equation="1-\alpha" contenteditable="false"><span></span><span></span></span> 置信区间，<span class="equation-text" data-index="11" data-equation="\theta_1和\theta_2" contenteditable="false"><span></span><span></span></span>分别成为<font color="#B71C1C">置信下限和置信上限</font><br>
正态总体参数的区间估计<br>
假设检验<br>
假设检验<br>
假设检验<br>
小概率原理<br>
原假设与备择假设<br>
两类错误
第一类：弃真错误
第二类：存伪错误
显著性检验<br>
拒绝域<br>
正态分布<br>
t 分布<br>
<span class="equation-text" data-index="0" data-equation="\mu " contenteditable="false"><span></span><span></span></span>分布<br>
<span class="equation-text" data-index="0" data-equation="\chi " contenteditable="false"><span></span><span></span></span>分布<br>
检验水平
在假设检验中允许犯第一类错误的概率，记为<span class="equation-text" data-index="0" data-equation="\alpha(0<\alpha<1)," contenteditable="false"><span></span><span></span></span>称 <span class="equation-text" data-index="1" data-equation="\alpha" contenteditable="false"><span></span><span></span></span>为检验水平<br>其表示对 <span class="equation-text" data-index="2" data-equation="H_0 " contenteditable="false"><span></span><span></span></span>弃真的程度，一般<span class="equation-text" data-index="3" data-equation=" \alpha" contenteditable="false"><span></span><span></span></span> 取 <span class="equation-text" data-index="4" data-equation="0.1,0.05,0.01,0.001" contenteditable="false"><span></span><span></span></span> 等<br>
显著性检验
只控制第一类错误概率 <span class="equation-text" data-index="0" data-equation="\alpha" contenteditable="false"><span></span><span></span></span> 的统计检验<br>
一般步骤
根据问题提出原假设 <span class="equation-text" data-index="0" data-equation="H_0" contenteditable="false"><span></span><span></span></span><br>
给出显著性水平 <span class="equation-text" data-index="0" data-equation="\alpha" contenteditable="false"><span></span><span></span></span><br>
确定检验统计量以及拒绝域形式<br>
按犯第一类错误的概率等于 <span class="equation-text" data-index="0" data-equation="\alpha" contenteditable="false"><span></span><span></span></span> 求出拒绝域 <span class="equation-text" data-index="1" data-equation="W" contenteditable="false"><span></span><span></span></span><br>
根据样本值计算检验统计量 <span class="equation-text" data-index="0" data-equation="T" contenteditable="false"><span></span><span></span></span> 的观测值 <span class="equation-text" data-index="1" data-equation="t," contenteditable="false"><span></span><span></span></span> 当 <span class="equation-text" data-index="2" data-equation="t\in W" contenteditable="false"><span></span><span></span></span>时，拒绝原假设 <span class="equation-text" data-index="3" data-equation="H_0" contenteditable="false"><span></span><span></span></span> 成立；否则，接受原假设 <span class="equation-text" data-index="4" data-equation="H_0" contenteditable="false"><span></span><span></span></span><br>
前置
条件关系<br>
<br>充要条件<br><span class="equation-text" data-index="0" data-equation="A 的充分条件是 B,B = A 的充分条件,B \implies A" contenteditable="false"><span></span><span></span></span>
集合关系<br>
二项式性质<br>
<span class="equation-text" data-index="0" data-equation="\sum_{i=0}^n C(n,i) = 2^n" contenteditable="false"><span></span><span></span></span>
随机事件及概率
概念<br>
随机实验
相同条件，实验可重复进行
所有可能结果已知<br>
实验具体结果未知<br>
样本点<br>
样本空间<br>
事件
类型<br>
随机事件
必然事件<br>
不可能事件
事件关系
子事件<br><span class="equation-text" data-index="0" data-equation="A\subset B" contenteditable="false"><span></span><span></span></span><br>
子集合关系【吸收律】<br><span class="equation-text" data-index="0" data-equation="A \subset B \iff AB=A \iff \\A \cup B=B\iff A\bar B= \varnothing \iff \bar B \subset \bar A" contenteditable="false"><span></span><span></span></span>
子事件<br><span class="equation-text" data-index="0" data-equation="A \subset B,P(A) \leq P(B)" contenteditable="false"><span></span><span></span></span><br>
相等<br><span class="equation-text" data-index="0" data-equation="A=B" contenteditable="false"><span></span><span></span></span><br>
并事件<br><span class="equation-text" data-index="0" data-equation="A \cup B" contenteditable="false"><span></span><span></span></span><br>
交事件<br><span class="equation-text" data-index="0" data-equation="A \cap B=AB" contenteditable="false"><span></span><span></span></span>
差事件<br><span class="equation-text" data-index="0" data-equation="A-B=A-AB=A\bar B" contenteditable="false"><span></span><span></span></span><br>
互斥事件<br><span class="equation-text" data-index="0" data-equation="A\cap B=\varnothing" contenteditable="false"><span></span><span></span></span>
对立事件<br><span class="equation-text" data-index="0" data-equation="A\cap B=\varnothing且A\cup B=\Omega,记为\bar{A}=B" contenteditable="false"><span></span><span></span></span><br>
事件运算
交换律<br><span class="equation-text" data-index="0" data-equation="AB=BA, A\cup B = B \cup A" contenteditable="false"><span></span><span></span></span><br>
结合律<br><span class="equation-text" data-index="0" data-equation="(AB)C=A(BC), (A\cup B)\cup C=A\cup (B\cup C)" contenteditable="false"><span></span><span></span></span>
吸收率<br><span class="equation-text" data-index="0" data-equation="A\sub B,AB=A,A\cup B=B" contenteditable="false"><span></span><span></span></span><br>
分配律<br><span class="equation-text" data-index="0" data-equation="(A\cup B)\cap C=(A\cap C)\cup(B\cap C) \\(A\cap B)\cup C=(A\cup C)\cap(B\cup C)" contenteditable="false"><span></span><span></span></span><br>
德摩根定理【对偶率】<br><span class="equation-text" data-index="0" data-equation="\overline{AB}=\overline{A}\cup \overline{B}，\overline{A\cup B}=\overline{A} \space\overline{B}" contenteditable="false"><span></span><span></span></span><br>
事件表示
假设表示<br>设事件A表示为...
表示事件/表示变量<br>设A<span class="equation-text" data-index="0" data-equation="=\{" contenteditable="false"><span></span><span></span></span>某某事件<span class="equation-text" data-index="1" data-equation="\};" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="2" data-equation="A_i=" contenteditable="false"><span></span><span></span></span>"由i次发生",<span class="equation-text" data-index="3" data-equation="i=..." contenteditable="false"><span></span><span></span></span>【条件概率】记某某事件A发生次数为X【概型】
集合表示<span class="equation-text" data-index="0" data-equation="A_i=\{" contenteditable="false"><span></span><span></span></span>某事件<span class="equation-text" data-index="1" data-equation="i\};A=\{(x,y)|f(x,y)>0,(x,y)\in \Omega\}" contenteditable="false"><span></span><span></span></span>
概率<br>
公理<br>
非负性<br>
规范性<br>
可列可加性<br>
性质/公式
规范性/非负性<br><span class="equation-text" data-index="0" data-equation="P(\varnothing)=0，P(\Omega)=1,0\leq P(A) \leq 1" contenteditable="false"><span></span><span></span></span><br>
两两互不相容事件可加【可列可加性】<br><span class="equation-text" data-index="0" data-equation="P(A_1\cup ... \cup A_n)=P(A_1)+...+P(A_n)" contenteditable="false"><span></span><span></span></span><br>
求逆公式【对立事件】<br><span class="equation-text" data-index="0" data-equation="P(\bar A)=P(\Omega)-P(A)=1-P(A)" contenteditable="false"><span></span><span></span></span><br>
减法公式【差事件】<br><span class="equation-text" data-index="0" data-equation="P(A-B)=P(A)-P(AB)=P(A\bar B)" contenteditable="false"><span></span><span></span></span><br>
加法公式【容斥定义】<br><span class="equation-text" data-index="0" data-equation="P(A\cup B)=P(A)+P(B)-P(AB)" contenteditable="false"><span></span><span></span></span><br>
三事件加法【三容斥】<br><span class="equation-text" data-index="0" data-equation="P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(AB)-P(AC)-P(BC)+P(ABC)=1-P(\bar A\bar B\bar C)" contenteditable="false"><span></span><span></span></span>
吸收率<br><span class="equation-text" data-index="0" data-equation="P(ABC \cup AB) = P(ABC)" contenteditable="false"><span></span><span></span></span><br><span class="equation-text" data-index="1" data-equation="P(AB\cap B)=P(AB)" contenteditable="false"><span></span><span></span></span>
德摩根定律<br><span class="equation-text" data-index="0" data-equation=" P(\bar A \bar B)= P(\bar{A \cup B})=1- P(A \cup B)" contenteditable="false"><span></span><span></span></span>
概率不等式<br><span class="equation-text" data-index="0" data-equation="P(A)\geq P(AB )" contenteditable="false"><span></span><span></span></span><br>
概率与事件关系<br>
<br><span class="equation-text" data-index="0" data-equation="A=B \implies P(A)=P(B)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="A=\varnothing \implies P(A)=0" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="A,B互斥 \implies P(AB)=0" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="B=\Omega \implies P(B)=1" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="A \subset B \implies P(A)=P(AB)" contenteditable="false"><span></span><span></span></span>
概型<br>
古典概型<br>
定义<br>有限样本点，且样本点发生可能性<b>相同</b>
公式<br><span class="equation-text" data-index="0" data-equation="P(A)=m/n, m为基本事件数，n为样本空间事件总数" contenteditable="false"><span></span><span></span></span>
推论【N件产品有M件次品，取n件，恰有k件次品概率】<br><span class="equation-text" data-index="0" data-equation="C_M^kC_{N-M}^{n-k}/C_N^n" contenteditable="false"><span></span><span></span></span><br>
加法/乘法原理<br>
加法即独立方法
乘法即步骤顺序<br>
排列/组和
排列<br><span class="equation-text" data-index="0" data-equation="A(n,m)=A_n^m=(n-m+1)...(n-1)n=n!/(n-m)!" contenteditable="false"><span></span><span></span></span><br>
组合<br><span class="equation-text" data-index="0" data-equation="C(n,m)=C_n^m=\frac{n!}{m!(n-m)!}=A_n^m/m!" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="C(n,m)=C(n-1,m-1)+C(n-1,m) " contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\sum_{i=0}^k C(k,i)=2^k" contenteditable="false"><span></span><span></span></span>
阶乘<br><span class="equation-text" data-index="0" data-equation="n!=1\times 2 \times ... \times n" contenteditable="false"><span></span><span></span></span><br>
几何概型<br>
定义<br>样本空间中无限样本点构成的<b>几何区域(长度、面积、体积等)</b>，且样本点等可能发生<br>
公式<br><span class="equation-text" data-index="0" data-equation="P(A)=L(A)/L(\Omega)" contenteditable="false"><span></span><span></span></span>
条件概率/乘法公式<br>
定义<br>已知A发生条件下B发生的概率，记为P(B|A)<br>
公式<br><span class="equation-text" data-index="0" data-equation="P(B|A)=P(AB)/P(A)" contenteditable="false"><span></span><span></span></span><br>
常用性质【前提P(B)>0】<br>
[0,1]范围<br><span class="equation-text" data-index="0" data-equation="0\leq P(A|B)\leq 1" contenteditable="false"><span></span><span></span></span><br>
规范性<br><span class="equation-text" data-index="0" data-equation="P(\varnothing|B)=0,P(\Omega|B)=1" contenteditable="false"><span></span><span></span></span><br>
求逆<br><span class="equation-text" data-index="0" data-equation="P(A|B)=1-P(\bar A|B)" contenteditable="false"><span></span><span></span></span><br>
注意: 不是 <span class="equation-text" data-index="0" data-equation="P(A|B)=1-P(A|\bar B)" contenteditable="false"><span></span><span></span></span><br>
加法公式【容斥性】<br><span class="equation-text" data-index="0" data-equation="P(A_1\cup A_2| B)=P(A_1|B)+P(A_2|B)-P(A_1A_2|B)" contenteditable="false"><span></span><span></span></span><br>
减法公式<br><span class="equation-text" data-index="0" data-equation="P(A_1-A_2|B)=P(A_1|B)-P(A_2|B)=P(A_1\bar A_2|B)" contenteditable="false"><span></span><span></span></span><br>
如果A,B独立<br><span class="equation-text" data-index="0" data-equation="P(AB)=P(A)P(B),P(A|B)=P(AB)/P(B)=P(A)" contenteditable="false"><span></span><span></span></span><br>
可列可加性
乘法公式<br>
<br><span class="equation-text" data-index="0" data-equation="【P(AB)】\underrightarrow{P(A)>0}【P(A)P(B|A)】\underrightarrow{P(B)>0}【P(B)P(A|B)】" contenteditable="false"><span></span><span></span></span>
全概率公式/贝叶斯公式<br>
全概率公式【无数现实之和=真理】<br><span class="equation-text" data-index="0" data-equation="P(B)=\sum_{i=1}^n P(A_i)P(B|A_i)=\sum_{i=1}^nP(A_iB)=P(B)\sum_{i=1}^nP(A_i)" contenteditable="false"><span></span><span></span></span><br>
贝叶斯公式【已知结果求路径概率】<br><span class="equation-text" data-index="0" data-equation="P(A_i|B)=P(A_iB)/P(B)=P(A_i)P(B|A_i)/P(B)" contenteditable="false"><span></span><span></span></span><br>
结论【抓阄原理】<br><span class="equation-text" data-index="0" data-equation="n个人抓m个阄，则第k个人抓到的概率为m/n" contenteditable="false"><span></span><span></span></span><br>
完备事件组<br><span class="equation-text" data-index="0" data-equation="1)A_1\cup A_2\cup...\cup A_n=\Omega,P(A_i)>0 \\ 2) A_i \cap A_j=\empty(i\neq j)" contenteditable="false"><span></span><span></span></span><br>
伯努利概型<br>
定义<br><span class="equation-text" data-index="0" data-equation="只有两个结果A,\bar A" contenteditable="false"><span></span><span></span></span><br>
n重伯努利实验<br><span class="equation-text" data-index="0" data-equation="C_n^kp^k(1-p)^{n-k},k=0,1,2,..,n" " contenteditable="false"><span></span><span></span></span><br>
事件独立性<br>
定义
独立定义<br><span class="equation-text" data-index="0" data-equation="P(AB)=P(A)P(B)\iff P(A|B)=P(A) \iff P(B|A)=P(B)" contenteditable="false"><span></span><span></span></span><br>
互斥定义<br><span class="equation-text" data-index="0" data-equation="A,B互斥\iff AB=\varnothing \implies P(AB)=0" contenteditable="false"><span></span><span></span></span><br>
重要结论<br>
<span class="equation-text" data-index="0" data-equation="A" contenteditable="false"><span></span><span></span></span> 与 <span class="equation-text" data-index="1" data-equation="B" contenteditable="false"><span></span><span></span></span> 独立，则 <span class="equation-text" data-index="2" data-equation="\bar A" contenteditable="false"><span></span><span></span></span>与 <span class="equation-text" data-index="3" data-equation="\bar B" contenteditable="false"><span></span><span></span></span> 对立，<span class="equation-text" data-index="4" data-equation="A" contenteditable="false"><span></span><span></span></span> 与 <span class="equation-text" data-index="5" data-equation="B" contenteditable="false"><span></span><span></span></span> 对立，<span class="equation-text" data-index="6" data-equation="\bar A" contenteditable="false"><span></span><span></span></span> 与 <span class="equation-text" data-index="7" data-equation="B" contenteditable="false"><span></span><span></span></span>，均相互独立<br>
概率为 0 的事件以及概率为 1 的事件与任一事件均相互独立<br>
独立条件下，条件概率只与子事件有关<br><span class="equation-text" data-index="0" data-equation="P(A|B)=P(A|\bar B)=P(A)\iff P(A|B)+P(\bar A|\bar B)=1 \iff P(A|\bar B)+P(\bar A|B)=1" contenteditable="false"><span></span><span></span></span>
A,B,C 两两独立<br><span class="equation-text" data-index="0" data-equation="A,B,C两两独立 \iff \begin{cases}P(AB)=P(A)P(B)\\P(AC)=P(A)P(C)\\P(BC)=P(B)P(C)\end{cases}" contenteditable="false"><span></span><span></span></span>
A,B,C 相互独立<br><span class="equation-text" data-index="0" data-equation="A,B,C相互独立 \iff \begin{cases}P(AB)=P(A)P(B)\\P(AC)=P(A)P(C)\\P(BC)=P(B)P(C)\\P(ABC)=P(A)P(B)P(C)\end{cases}" contenteditable="false"><span></span><span></span></span>
设有 n(n>=2) 个随机事件 A1,...,Am ,如果对其中任 k 个，均有<br><span class="equation-text" data-index="0" data-equation="P(A_{i1}...A_{ik})=P(A_{i1})..P(A_{in})，则称事件组独立" contenteditable="false"><span></span><span></span></span><br>
事件独立性标志<br>"互不干扰"，"互不影响","有放回取球"<br>
独立互斥关系<br>
没有直接联系<br>
独立重复实验
<br><span class="equation-text" data-index="0" data-equation="C_n^kp^k(1-p)^{n-k},k=0,1,2,..,n" contenteditable="false"><span></span><span></span></span>
独立重复实验，"第n次实验刚好是事件第k次发生"<br><span class="equation-text" data-index="0" data-equation="C_{n-1}^{k-1} p^{k-1}(1-p)^{n-k}\cdot p=C_{n-1}^{k} p^{k-1}(1-p)^{n-k}" contenteditable="false"><span></span><span></span></span><br>
一维随机变量及其分布<br>
随机变量
定义<br><span class="equation-text" data-index="0" data-equation="样本空间上的单值实值函数X=X(\omega),\omega为样本点" contenteditable="false"><span></span><span></span></span>
事件表示<br><span class="equation-text" data-index="0" data-equation="硬币实验\Omega=\{正面，反面\}=\{H,T\},X=\begin{cases}0,\omega=H\\1,\omega=T\end{cases}" contenteditable="false"><span></span><span></span></span><br>
分布函数
定义<br>
<span class="equation-text" data-index="0" data-equation="F(x)=P\{X\leq x\},x\in (-\infty,+\infty)为随机变量X的分布函数" contenteditable="false"><span></span><span></span></span>
性质
域 / 规范性<br><span class="equation-text" data-index="0" data-equation="0\leq F(x)\leq 1,F(-\infty)=0,F(+\infty)=1" contenteditable="false"><span></span><span></span></span>
单增<br><span class="equation-text" data-index="0" data-equation="F(x)是单调不减的函数，即对任意x_1<x_2,均有F(x_1)\leq F(x_2)" contenteditable="false"><span></span><span></span></span>
右连续<br><span class="equation-text" data-index="0" data-equation="F(x)处处右连续，对任意x_0有,F(x_0+0)=F(x_0)=\lim_{x\rightarrow x_0^+}F(x)【a< x\leq b】" contenteditable="false"><span></span><span></span></span>
计算<br>
基本公式<br>
<span class="equation-text" data-index="0" data-equation="P\{X=x\}=F(x)-F(x-0)=F(x)-\lim_{x\rightarrow x^-}F(x)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="P\{X>x\}=1-F(x)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="P\{X\geq x\}=1-F(x-0)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="P\{a<X\leq b\}=F(b)-F(a)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="P\{a<X<b\}=F(b-0)-F(a)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="=P\{a<X\leq b\}-P\{X=b\}=F(b)-F(a)-[F(b)-F(b-0)]" contenteditable="false"><span></span><span></span></span>
反函数<br>
<span class="equation-text" data-index="0" data-equation="随机变量X分布函数为F(x)，则Y=f(x)的分布函数G(y)=F(f^{-1}(y))" contenteditable="false"><span></span><span></span></span>
复合运算<br>
分布函数的线性组合仍为分布函数【系数和为1，保证概率规范性】<br><span class="equation-text" data-index="0" data-equation="1.a_i \geq 0,a_1+a_2=1时，a_1F_1(x)+a_2F_2(x)仍为分布函数" contenteditable="false"><span></span><span></span></span>
分布函数的乘积<br><span class="equation-text" data-index="0" data-equation="2.F_1(x)F_2(x)仍为分布函数" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="3.1-[1-F_1(x)][1-F_2(x)]" contenteditable="false"><span></span><span></span></span>
密度函数的复合运算<br><span class="equation-text" data-index="0" data-equation="a_1\geq0,a_2\geq 0,且a_1+a_2=1时,a_1f_1(x)+a_2f_2(x)必为密度函数" contenteditable="false"><span></span><span></span></span><br>
经典错误<br>
<br><span class="equation-text" data-index="0" data-equation="设X的分布函数为F(x),则1-F(-x)仍为分布函数" contenteditable="false"><span></span><span></span></span>
离散型随机变量
定义<br>
概率分布 / 分布律<br>
表格形式<br><span class="equation-text" data-index="0" data-equation="X\sim \begin{pmatrix}-1&0&1&2\\1/4&1/4&1/3&1/6\end{pmatrix}" contenteditable="false"><span></span><span></span></span>
公式形式<br><span class="equation-text" data-index="0" data-equation="P\{X=x_i\}=p_i,i=1,2,3..." contenteditable="false"><span></span><span></span></span><br>
分布律性质<br>
<br><span class="equation-text" data-index="0" data-equation="p_i \geq 0,i=1,2,3..." contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="\sum_i p_i=1" contenteditable="false"><span></span><span></span></span>
常见类型
0-1分布 <br><span class="equation-text" data-index="0" data-equation="X \sim B(1,p)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="P\{X=k\}=p^k(1-p)^{1-k},k=0,1\ " contenteditable="false"><span></span><span></span></span><br>
二项分布 Binomial Distribution<br><span class="equation-text" data-index="0" data-equation="X\sim B(n,p)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="P\{X=k\}=C_n^kp^k(1-p)^{n-k},k=0,1,2,..,n" contenteditable="false"><span></span><span></span></span><br>
泊松分布 Poisson Distribution<br><span class="equation-text" data-index="0" data-equation="X \sim P(\lambda)" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="P\{X=k\}=\frac{\lambda^k}{k!}e^{-\lambda},k=0,1,2,..." contenteditable="false"><span></span><span></span></span>
泊松定理
条件<br> <span class="equation-text" data-index="0" data-equation="\lim_{n\rightarrow \infty}C_n^kp_n^k(1-p_n)^{n-k}=\frac{\lambda^ke^{-\lambda}}{k!}" contenteditable="false"><span></span><span></span></span><br>
推理记忆<br>
级数<br><span class="equation-text" data-index="0" data-equation="\sum_{k=0}^n\frac{\lambda^k}{k!}=1+\lambda+\frac{\lambda^2}{2!}+...+\frac{\lambda^n}{n!}=e^{\lambda}" contenteditable="false"><span></span><span></span></span>
期望<br><span class="equation-text" data-index="0" data-equation="E(x) = \int xf(x)dx=\sum \frac{\lambda \times \lambda^{x-1}}{x-1}e^{-\lambda}=\lambda \times 1 = \lambda" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="E(X)=\lambda=np,p=\frac{\lambda}{n},P(k)=\lim_{n\rightarrow \infty}C_n^k\cdot \frac{\lambda^k}{n}(1-\frac{\lambda}{n})^{n-k}=\frac{\lambda^k}{n}[(1+1/-\frac{n}\lambda)^{-\frac{n}{\lambda}}]^{-\lambda}=\frac{\lambda^k}{k!}e^{-\lambda}" contenteditable="false"><span></span><span></span></span>
高数结论【级数】<br><span class="equation-text" data-index="0" data-equation="\sum \frac{1}{\lambda^k \cdot k!}=\sum \frac{(\frac{1}{\lambda})^k}{k!}=e^{\frac{1}{\lambda}}" contenteditable="false"><span></span><span></span></span>
几何分布<br><span class="equation-text" data-index="0" data-equation="X \sim GE(p)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="P\{X=k\}=(1-p)^{k-1}p,k=0,1..." contenteditable="false"><span></span><span></span></span>
超几何分布<br><span class="equation-text" data-index="0" data-equation="X\sim H(n,M,N)" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="P\{X=k\}=C_M^kC_{N-M}^{n-k}/C_N^n" contenteditable="false"><span></span><span></span></span>
连续型随机变量<br>
定义<br>
<br><span class="equation-text" data-index="0" data-equation="随机变量X分布函数为F(x),若存在非负可积f(x)\\对任意实数x，有F(x)=\int_{-\infty}^x f(t)dt,即称f(x)为X的概率密度函数" contenteditable="false"><span></span><span></span></span>
连续性<br>分布函数不连续的随机变量，一定不是连续性随机变量<br>
概率密度性质<br>
非负性<br><span class="equation-text" data-index="0" data-equation="f(x)\geq 0" contenteditable="false"><span></span><span></span></span>
规范性<br><span class="equation-text" data-index="0" data-equation="\int_{-\infty}^{+\infty}f(x)dx=1" contenteditable="false"><span></span><span></span></span>
概率密度为偶函数<br><span class="equation-text" data-index="0" data-equation="F(x)+F(-x)=1,F(x)一定不为奇函数【不存在小于零部分】" contenteditable="false"><span></span><span></span></span><br>
f(x)连续点处<br><span class="equation-text" data-index="0" data-equation="F'(x)=f(x)" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="x_1<x_2,有P\{x_1<X\leq x_2\}=\int_{x_1}^{x_2} f(t)dt" contenteditable="false"><span></span><span></span></span>
分布函数
概率为0【极限】事件未必为不可能事件，但不可能事件概率必定为0<br><span class="equation-text" data-index="0" data-equation="\forall x_0\in(-\infty,+\infty),P\{X=x_0\}=F(x_0)-F(x_0-0)=0" contenteditable="false"><span></span><span></span></span>
事件<font color="#B71C1C">端点不会影响</font>区间概率【注意与离散型随机变量之间的区分】<br><span class="equation-text" data-index="0" data-equation="P\{a<X\leq b\}=P\{a\leq X\leq b\}=P\{a\leq X< b\}=P\{a<X< b\}=F(b)-F(a)=\int_a^bf(x)dx" contenteditable="false"><span></span><span></span></span>
常见类型<br>
均匀分布 Uniform Distribution<br><span class="equation-text" data-index="0" data-equation="X\sim U(a,b)" contenteditable="false"><span></span><span></span></span><br>
概率密度<br><span class="equation-text" data-index="0" data-equation="f(x)=\begin{cases}1/(b-a),a \leq x \leq b \\0,其他\end{cases}" contenteditable="false"><span></span><span></span></span><br>
分布函数<br><span class="equation-text" data-index="0" data-equation="F(x)=\begin{cases}0,x<a \\ (x-a)/(b-a),a\leq x < b\\ 1,x\geq b\\\end{cases}" contenteditable="false"><span></span><span></span></span><br>
对称轴<br><span class="equation-text" data-index="0" data-equation="P\{X>\frac{a+b}{2}\}=P\{X<\frac{a+b}{2}\}=\frac{1}{2}" contenteditable="false"><span></span><span></span></span>
区间概率<br><span class="equation-text" data-index="0" data-equation="[c,d]\subset(a,b),P\{c<X<d\}=\frac{d-c}{b-a}" contenteditable="false"><span></span><span></span></span>
指数分布 Exponential Distribution<br><span class="equation-text" data-index="0" data-equation="X \sim E(\lambda)" contenteditable="false"><span></span><span></span></span><br>
概率密度<br><span class="equation-text" data-index="0" data-equation="f(x)=\begin{cases}\lambda e^{-\lambda x} , x>0 \\0,x\leq0\end{cases}" contenteditable="false"><span></span><span></span></span><br>
分布函数<br><span class="equation-text" data-index="0" data-equation="F(x)=\begin{cases}0, x<0 \\1-e^{-\lambda x} ,x\geq 0\end{cases}" contenteditable="false"><span></span><span></span></span><br>
证明:<br><span class="equation-text" data-index="0" data-equation="P\{x\leq X\leq s+x|X>x\}/h=\lambda" contenteditable="false"><span></span><span></span></span><br>
假设分布函数 <span class="equation-text" data-index="0" data-equation="F(x)=P\{X\leq x\}" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="\lim_{h\rightarrow 0} [F(x+h)-F(x)]/h(1-F(x))=\lim_{h\rightarrow 0} F'(x)/(1-F(x))=\lambda" contenteditable="false"><span></span><span></span></span>
化简为 <span class="equation-text" data-index="0" data-equation="\frac{dF(x)}{dx}+\lambda F(x)=\lambda" contenteditable="false"><span></span><span></span></span><br>
求解 <span class="equation-text" data-index="0" data-equation="F(x)=[F(0)-\frac{\lambda}{\lambda}]e^{-\lambda x}+\frac{\lambda}{\lambda}=1-e^{\lambda x}" contenteditable="false"><span></span><span></span></span>
无记忆性【区间长度相等概率相同】<br><span class="equation-text" data-index="0" data-equation="P\{X>s+t|X>s\}=P\{X>t\}=e^{-\lambda t}" contenteditable="false"><span></span><span></span></span><br>
正态分布 normal distribution<br><span class="equation-text" data-index="0" data-equation="X \sim N(\mu,\sigma^2)" contenteditable="false"><span></span><span></span></span><br>
密度函数<br><span class="equation-text" data-index="0" data-equation="f(x)= \frac{1}{\sqrt{2\pi} \sigma}\times e^{-\frac{(x-\mu)^2}{2\sigma^2}}" contenteditable="false"><span></span><span></span></span><br>
分布函数<br><span class="equation-text" data-index="0" data-equation="F(x)=\int_{-\infty}^x f(x)dx=f(x)=\int_{-\infty}^x 1/(\sqrt{2\pi}\sigma)*e^{-\frac{(x-\mu)^2}{2\sigma^2}}" contenteditable="false"><span></span><span></span></span><br>
性质<br>
标准态分布<br><span class="equation-text" data-index="0" data-equation="X\sim N(0,1)" contenteditable="false"><span></span><span></span></span><br>
概率密度<br><span class="equation-text" data-index="0" data-equation="\varphi(x)= 1/(\sqrt{2\pi})*e^{-\frac{x^2}{2}}" contenteditable="false"><span></span><span></span></span>
分布函数<br><span class="equation-text" data-index="0" data-equation="\Phi(x)=\int_{-\infty}^x f(x)dx" contenteditable="false"><span></span><span></span></span><br>
分布函数关于X=0对称<br><span class="equation-text" data-index="0" data-equation="\Phi(-x)=1-\Phi(x)" contenteditable="false"><span></span><span></span></span>
利用标准正态分布求解<br><span class="equation-text" data-index="0" data-equation="X \sim N(\mu,\sigma^2)" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="P\{X=k\}=A/(2^k\cdot k!),求A" contenteditable="false"><span></span><span></span></span>
概率区间标准化求解<br><span class="equation-text" data-index="0" data-equation="P\{a<X\leq b\}=P\{\frac{a-\mu}{\sigma}<\frac{X-\mu}{\sigma}\leq \frac{b-\mu}{\sigma}\}=\Phi(\frac{b-\mu}{\sigma})-\Phi(\frac{a-\mu}{\sigma})" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="P\{X\leq x\}=P\{\frac{X-\mu}{\sigma}<\frac{x-\mu}{\sigma}\}=\Phi(\frac{x-\mu}{\sigma})" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="F(x)=C_1\Phi(\frac{x-\mu_1}{\sigma})+C_2\Phi(\frac{a-\mu_2}{\sigma}),C_1+C_2=1,\implies EX=C_1\mu_1+C_2\mu_2" contenteditable="false"><span></span><span></span></span>
图形性质<br>
对称轴<br><span class="equation-text" data-index="0" data-equation="X=\mu" contenteditable="false"><span></span><span></span></span><br>
对称性<br><span class="equation-text" data-index="0" data-equation="P\{X>\mu\}=P\{X<\mu\}=\frac{1}{2}" contenteditable="false"><span></span><span></span></span><br>
离散程度<br><span class="equation-text" data-index="0" data-equation="\sigma" contenteditable="false"><span></span><span></span></span> 越小，分布越集中，图越高越瘦，反之越分散，又矮又胖
标准正态<br><span class="equation-text" data-index="0" data-equation="X\sim N(0,1)" contenteditable="false"><span></span><span></span></span>
即位于U_α 右侧概率面积为 α<br><span class="equation-text" data-index="0" data-equation="U\sim N(0,1)，对于\alpha,0<\alpha<1,称满足P\{U\geq U_\alpha\}=\alpha" contenteditable="false"><span></span><span></span></span>
复合变量【<font color="#B71C1C">原理：方差和均值</font>公式】<br><span class="equation-text" data-index="0" data-equation="Y=ax+b\sim N(a\mu+b,a^2\sigma^2),a\neq 0" contenteditable="false"><span></span><span></span></span><br>
高数结论<br><span class="equation-text" data-index="0" data-equation="\int_0^{+\infty} e^{-x^2}dx=\frac{1}{2}\int_{-\infty}^{+\infty}e^{-x^2}dx=\frac{1}{2}\sqrt{x}【\mu=0,\sigma=1/\sqrt{2}】" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="f(x)=Ce^{-x^2+2x}求C" contenteditable="false"><span></span><span></span></span>
随机变量函数分布
离散型
利用分布律求解<br><span class="equation-text" data-index="0" data-equation="P\{Y=g(x_k)\} = p_k,k=1,2..." contenteditable="false"><span></span><span></span></span><br>
连续型
公式法<br><span class="equation-text" data-index="0" data-equation="f_Y(y)=\begin{cases} |h'(y)|f_X(h(y)),\alpha<y<\beta \\0,其他 \end{cases}" contenteditable="false"><span></span><span></span></span><br>
定义法<br><span class="equation-text" data-index="0" data-equation="F_Y(y)=P\{Y\leq y\}=P\{g(X)\leq y\}=P\{\psi(y)\leq X\leq \varphi(y)\}" contenteditable="false"><span></span><span></span></span>
通用型<br>
分布函数的线性组合仍为分布函数【系数和为1，保证概率规范性】<br><span class="equation-text" data-index="0" data-equation="1.a_i \geq 0,a_1+a_2=1时，a_1F_1(x)+a_2F_2(x)仍为分布函数" contenteditable="false"><span></span><span></span></span>
分布函数的乘积<br><span class="equation-text" data-index="0" data-equation="F_1(x)F_2(x)仍为分布函数 \\ X ,Y独立时，max\{X,Y\}的分布函数" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="1-[1-F_1(x)][1-F_2(x)]\\X ,Y独立时，min\{X,Y\}的分布函数" contenteditable="false"><span></span><span></span></span>
密度函数的复合运算<br><span class="equation-text" data-index="0" data-equation="a_1\geq0,a_2\geq 0,且a_1+a_2=1时,a_1f_1(x)+a_2f_2(x)必为密度函数" contenteditable="false"><span></span><span></span></span><br>
常见题型<br>
练习<br>
分布函数/概率密度基本概念<br>
分布中待定参数求解
一维连续型随机变量<br>
常见分布
函数的分布<br>
真题<br>
二维随机变量及其分布<br>
定义<br>
n 维随机向量<br><span class="equation-text" data-index="0" data-equation="(X_1,X_2,...,X_n)" contenteditable="false"><span></span><span></span></span><br>
【(X,Y)的联合分布函数】<br><span class="equation-text" data-index="0" data-equation="F(x,y)=P\{(X\leq x)\cap(Y\leq y)\}=P\{X\leq x,Y\leq y\},x,y\in (-\infty,+\infty)" contenteditable="false"><span></span><span></span></span>
边缘分布<br><span class="equation-text" data-index="0" data-equation="F_X(x)=P\{X\leq x\}=P\{X\leq x,Y<+\infty \}=F(x,+\infty)" contenteditable="false"><span></span><span></span></span><br>
条件分布<br><span class="equation-text" data-index="0" data-equation="\forall \xi>0,P\{y-\xi<Y\leq y+\xi\}>0,则\lim_{\xi\rightarrow 0^+}P\{X\leq x|y-\xi<Y\leq y+\xi\}=\lim_{\xi\rightarrow 0^+}\frac{P\{X\leq x|y-\xi<Y\leq y+\xi\}}{P\{y-\xi<Y\leq y+\xi\}},记作F_{X|Y}(x|y)" contenteditable="false"><span></span><span></span></span><br>
性质
边界 / 规范性<br><span class="equation-text" data-index="0" data-equation="0\leq F(x,y) \leq 1,有 F(-\infty,-\infty)=F(-\infty,y)=F(x,-\infty)=0,F(+\infty,+\infty)=1" contenteditable="false"><span></span><span></span></span>
单调性<br><span class="equation-text" data-index="0" data-equation="F(x_1,y)\leq F(x_2,y),F(x,y_1)\leq F(x,y_2)【x_2>x_1,y_2>y_1】" contenteditable="false"><span></span><span></span></span><br>
右连续<br><span class="equation-text" data-index="0" data-equation="F(x+0,y)=F(x,y),F(x,y+0)=F(x,y)" contenteditable="false"><span></span><span></span></span>
面积差分公式<br><span class="equation-text" data-index="0" data-equation="P\{x_1< X \leq x_2,y_1<Y\leq y_2\}=F(x_2,y_2)-F(x_2,y_1)-F(x_1,y_2)+F(x_1,y_1)\geq 0" contenteditable="false"><span></span><span></span></span><br>
离散型随机变量<br>
概率(联合)分布律<br>
性质<br>
<br><span class="equation-text" data-index="0" data-equation="p_{ij} \geq 0,i=1,2,3..." contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="P\{(X,Y)\in D\}=\sum _i\sum_j p_{ij}=1" contenteditable="false"><span></span><span></span></span>
书写格式<br>
矩阵法
枚举法<br>
边缘分布律
性质<br><span class="equation-text" data-index="0" data-equation="P\{X=x_i\}=P\{X=x,Y<+\infty\}=\sum_j p_{ij}=p_i" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="P\{Y=y_j\}=P\{X<+\infty,Y<y_j\}=\sum_j p_{ij}=p_j" contenteditable="false"><span></span><span></span></span>
条件分布律<br>
<br><span class="equation-text" data-index="0" data-equation="P\{X=x_i|Y=y_j\}=p_{ij}/p_{j}" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="(X|Y=y_i)" contenteditable="false"><span></span><span></span></span>
连续性随机变量<br>
联合分布函数
<br><span class="equation-text" data-index="0" data-equation="F(x,y)=\int_{-\infty}^x \int_{-\infty}^y f(u,v)dudv" contenteditable="false"><span></span><span></span></span>
联合概率密度<br>
非负性<br><span class="equation-text" data-index="0" data-equation="f(x,y)\geq 0" contenteditable="false"><span></span><span></span></span>
规范性<br><span class="equation-text" data-index="0" data-equation="\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}f(x,y)dxdy=1" contenteditable="false"><span></span><span></span></span>
连续点处概率密度 = 分布函数二阶混合偏导<br><span class="equation-text" data-index="0" data-equation="f(x,y)=\frac{\partial ^2F(x,y)}{\partial x\partial y}" contenteditable="false"><span></span><span></span></span><br>
计算<br>
设 D 为 xOy 平面区域，则点 (X,Y) 落在点 D 内的概率<br><span class="equation-text" data-index="0" data-equation="P\{(X,Y)\in D\}=\iint_Df(x,y)dxdy" contenteditable="false"><span></span><span></span></span><br>
边缘密度函数<br>
求X边缘密度，把X=x，对于的 y 积分成一根线的密度<br><span class="equation-text" data-index="0" data-equation="f_X(x)=\int_{-\infty}^{+\infty}f(x,y)dy,-\infty<x<+\infty" contenteditable="false"><span></span><span></span></span>
求Y边缘密度，把Y=y，对于的 x 积分成一根线的密度<br><span class="equation-text" data-index="0" data-equation="f_Y(y)=\int_{-\infty}^{+\infty}f(x,y)dx,-\infty<y<+\infty" contenteditable="false"><span></span><span></span></span>
条件概率密度<br>
<br><span class="equation-text" data-index="0" data-equation="Y=y(f_Y(y)>0)条件下，X的条件概率密度\\f_{X|Y}(x|y)=f(x,y)/f_Y(y),-\infty<x<+\infty" contenteditable="false"><span></span><span></span></span>
性质<br>
非负性<br>
规范性<br>
随机变量独立性<br>
定义<br><span class="equation-text" data-index="0" data-equation="F(x,y)=F_X(x)F_Y(y)" contenteditable="false"><span></span><span></span></span>
结论<br>
<span class="equation-text" data-index="0" data-equation="若X与Y相互独立，则X^2和Y^2相互独立，但反之未必" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="f(x,y)=\begin{cases}kg(x)h(y),a\leq x \leq b,c\leq y\leq d,\\0,其他\end{cases}，当1/k=I_1I_2=k\int_a^bg(x)dx\int_c^d h(y)dy时独立" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="P\{a<X\leq b,c< Y\leq d\}=P\{a<X\leq b\}P\{c<Y\leq d\}" contenteditable="false"><span></span><span></span></span>
判断
求出对应X,Y的边缘分布，再通过定义【充要条件】，进行判断
二维均匀分布<br><span class="equation-text" data-index="0" data-equation="(X,Y)\sim U(G)" contenteditable="false"><span></span><span></span></span>
概率密度<br><span class="equation-text" data-index="0" data-equation="f(x,y)=\begin{cases}1/A,(x,y)\in G \\0,其他\end{cases}" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="P\{(X,Y)\in D\}=P\{(X,Y)\in G\cap D\}=\frac{S_{G\cap D}}{S_G}" contenteditable="false"><span></span><span></span></span>
二维正态分布<br><span class="equation-text" data-index="0" data-equation="(X,Y)\sim N(\mu_1,\mu_2;\sigma_1^2,\sigma_2^2;\rho)" contenteditable="false"><span></span><span></span></span><br>
公式不需要记，但是要知道其对应的 期望/方差性质<br><span class="equation-text" data-index="0" data-equation="f(x,y)=1/(2\pi\sigma_1\sigma_2\sqrt{1-\rho^2})\exp\{...\} " contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="X与Y独立 \iff \rho(协方差) = 0" contenteditable="false"><span></span><span></span></span>
正态分布<br><span class="equation-text" data-index="0" data-equation="Z=aX+bY\sim N(a\mu_1+b\mu_2,a^2\sigma_1^2+b^2\sigma_2^2),a^2+b^2\neq 0" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="n个随机变量X_i相互独立，X_i\sim N(\mu,\sigma^2),有\sum_{i=1}^n c_iX_i\sim N(\sum c_i\mu,\sum c_j^2\sigma_j^2)" contenteditable="false"><span></span><span></span></span>
（X，Y）的正态分布性质<br>
（X,Y）二维正态<span class="equation-text" data-index="0" data-equation=" \implies " contenteditable="false"><span></span><span></span></span>则X,Y均正态<br>
(X,Y) 正态，当<span class="equation-text" data-index="0" data-equation="ad-bc \neq 0" contenteditable="false"><span></span><span></span></span>,则 (aX+bY,cX+dY) 也正态<br>
(X,Y) 正态 ，对任意常数 a 与 b, <span class="equation-text" data-index="0" data-equation="a^2+b^2\neq 0" contenteditable="false"><span></span><span></span></span> 时，aX+bY 必正态<br>
X,Y均正态且独立，则 <span class="equation-text" data-index="0" data-equation="(X,Y) \sim N(x,x;x,x;0)" contenteditable="false"><span></span><span></span></span><br>
简单函数分布<br>
二维离散型<br>
泊松分布<br><span class="equation-text" data-index="0" data-equation="X\sim P(\lambda_1),Y\sim P(\lambda_2),X+Y\sim P(\lambda_1+\lambda_2)" contenteditable="false"><span></span><span></span></span><br>
二项分布<br><span class="equation-text" data-index="0" data-equation="X\sim B(m,p),Y\sim B(n,p)，则X+Y\sim B(m+n,p)" contenteditable="false"><span></span><span></span></span><br>
二维连续型<br>
分布函数法<br><span class="equation-text" data-index="0" data-equation="Z=g(X,Y)分布函数为F_Z(z),自变量不一定是z" contenteditable="false"><span></span><span></span></span><br>
分布函数<br><span class="equation-text" data-index="0" data-equation="F_Z(z)=P\{Z\leq z\}=P\{g(X,Y)\leq z\}=\\ \iint_{g(x,y)\leq z}f(x,y)dxdy,-\infty <z< +\infty" contenteditable="false"><span></span><span></span></span>
概率密度<br><span class="equation-text" data-index="0" data-equation="f_Z(z)=F_Z'(z)" contenteditable="false"><span></span><span></span></span><br>
公式法
<span class="equation-text" data-index="0" data-equation="U=max\{X,Y\}" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="F_Z=F(x,x)【F(x,y)为(X,Y)的分布函数】" contenteditable="false"><span></span><span></span></span>
当 X,Y 独立时<br><span class="equation-text" data-index="0" data-equation="F_U=F_X(x)F_Y(x)" contenteditable="false"><span></span><span></span></span><br>
独立同分布<br><span class="equation-text" data-index="0" data-equation="F_X^2(x)【F_X=F_Y】" contenteditable="false"><span></span><span></span></span><br>
多维独立随机变量<br><span class="equation-text" data-index="0" data-equation="F_M(x)=P\{M\leq x\}=F_1(x)...F_n(x)" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="U=max\{X,Y\}=\frac{X+Y+|X-Y|}{2}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="V=min\{X,Y\}" contenteditable="false"><span></span><span></span></span><br>
<br><span class="equation-text" data-index="0" data-equation="F_V(v)=1-[1-F_X(v)][1-F_Y(v)]" contenteditable="false"><span></span><span></span></span>
X,Y 独立同分布时<br><span class="equation-text" data-index="0" data-equation="1-[1-F_X(x)^2]" contenteditable="false"><span></span><span></span></span><br>
指数独立同分布 X,Y ~ E(λ)<br><span class="equation-text" data-index="0" data-equation="min\{X,Y \}\sim E(\lambda_1+\lambda_2)" contenteditable="false"><span></span><span></span></span><br>
多维独立随机变量<br><span class="equation-text" data-index="0" data-equation="F_N(x)=P\{N\leq x\}=1-[1-F_1(x)]...[1-F_n(x)]" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="V=min\{X,Y\}=\frac{X+Y-|X-Y|}{2}" contenteditable="false"><span></span><span></span></span>
指数分布<br><span class="equation-text" data-index="0" data-equation="X\sim E(\lambda_1),Y\sim E(\lambda_2),则,\min\{X,Y\}\sim E(\lambda_1+\lambda_2)" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="Z=X+Y" contenteditable="false"><span></span><span></span></span>
<br><span class="equation-text" data-index="0" data-equation="f_Z(z)=\int_{-\infty}^{+\infty}f(x,z-x)dx 或 f_Z(z)=\int_{-\infty}^{+\infty}f(z-y,y)dx" contenteditable="false"><span></span><span></span></span><br>
X 和 Y 相互独立时【卷积公式】<br><span class="equation-text" data-index="0" data-equation="f_Z(z)=\int_{-\infty}^{+\infty}f_X(x)f_Y(z-x)dx \\或 f_Z(z)=\int_{-\infty}^{+\infty}f_X(z-y)f_Y(y)dx" contenteditable="false"><span></span><span></span></span><br>
正态分布<br><span class="equation-text" data-index="0" data-equation="X\sim N(\mu_1,\sigma_1^2),Y\sim N(\mu_2,\sigma^2),则·Z=aX\pm bY\sim N(a\mu_1\pm b\mu_2,a^2\sigma_1^2+b^2\sigma_2^2),a^2+b^2\neq 0" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="Z=Z(U,V)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="U+V" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="max\{X,Y\}+min\{X,Y\}=X+Y" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="UV" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="max\{X,Y\}min\{X,Y\}=XY" contenteditable="false"><span></span><span></span></span>
随机变量相互独立<br>
题型
练习<br>
求分布函数的参数<br>
已知X,Y的分布律，以及某关系，求其联合分布律<br>
真题
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