﻿ 关于条件分布与条件数字特征的研究

# 关于条件分布与条件数字特征的研究Research on the Conditional Distribution and the Conditional Digital Characteristics

Abstract:

Abstract: In recent years, with the continuous observation and study of random phenomena, conditional distribution and conditional expectation have been widely used in various fields, but there are still a lot of shortcomings in the existing theories. In this paper, the conditional distribution and its digital characteristics are studied. Not only the definition of classical conditional distribution and extended conditional distribution is introduced, but also the density function form and the distri-bution function form of the full probability formula are given. In addition, by discussing the digital features of the conditional distribution, we prove and generalize the new properties of the condi-tional expectation, for example, the full probability form of the heavy expectation formula, and also give the definition of the conditional variance and the proof of its properties.

1. 引言

2. 条件分布

2.1. 经典条件分布

$\left(X,Y\right)$ 是二维离散型随机变量，其联合分布列为

${p}_{ij}=P\left(X={x}_{i},Y={y}_{j}\right)$ , $i=1,2,\cdots$ , $j=1,2,\cdots$

${p}_{i|j}=P\left(X={x}_{i}|Y={y}_{j}\right)=\frac{P\left(X={x}_{i},Y={y}_{j}\right)}{P\left(Y={y}_{j}\right)}=\frac{{p}_{ij}}{{p}_{\cdot j}}$ , $i=1,2,\cdots$ (2.1.1)

$F\left(x|{y}_{j}\right)=\underset{{x}_{i}\le x}{\sum }P\left(X={x}_{i}|Y={y}_{j}\right)=\underset{{x}_{i}\le x}{\sum }{p}_{i|j}$ (2.1.2)

$\left(X,Y\right)$ 是二维连续型随机变量，其联合密度函数为 $f\left(x,y\right)$ ，边际密度函数为 ${f}_{X}\left(x\right)$${f}_{Y}\left(y\right)$

$F\left(x|y\right)={\int }_{-\infty }^{x}\frac{f\left(u,y\right)}{{f}_{Y}\left(y\right)}\text{d}u$ (2.1.3)

${f}_{X|Y}\left(x|y\right)=\frac{f\left(x,y\right)}{{f}_{Y}\left(y\right)}$ (2.1.4)

$f\left(x,y\right)={f}_{Y}\left(y\right){f}_{X|Y}\left(x|y\right)$ (2.1.5)

${f}_{X}\left(x\right)={\int }_{-\infty }^{\infty }{f}_{Y}\left(y\right){f}_{X|Y}\left(x|y\right)\text{d}y$ (2.1.6)

${f}_{Y|X}\left(y|x\right)=\frac{{f}_{Y}\left(y\right){f}_{X|Y}\left(x|y\right)}{{\int }_{-\infty }^{\infty }{f}_{Y}\left(y\right){f}_{X|Y}\left(x|y\right)\text{d}y}$ (2.1.7)

${F}_{X}\left(x\right)={\int }_{-\infty }^{\infty }F\left(x|y\right){f}_{Y}\left(y\right)\text{d}y$ (2.1.8)

${\int }_{-\infty }^{\infty }F\left(x|y\right){f}_{Y}\left(y\right)\text{d}y={\int }_{-\infty }^{\infty }\left({\int }_{-\infty }^{x}{f}_{X|Y}\left(x|y\right)\text{d}x\right){f}_{Y}\left(y\right)\text{d}y={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{x}f\left(x,y\right)\text{d}x\text{d}y={F}_{X}\left(x\right)$

2.2. 扩展条件分布

${F}_{X|B}\left(x|B\right)=P\left(X\le x|B\right)=P\left(A|B\right)=\frac{P\left(AB\right)}{P\left(B\right)}$ (2.2.1)

$\left(X,Y\right)$ 为二维连续型随机变量时，对一切使得 ${F}_{Y}\left(y\right)>0$ 的y，称

${F}_{X|Y\le y}\left(x|Y\le y\right)=P\left(X\le x|Y\le y\right)=\frac{P\left(X\le x,Y\le y\right)}{P\left(Y\le y\right)}=\frac{{\int }_{-\infty }^{x}{\int }_{-\infty }^{y}f\left(u,v\right)\text{d}u\text{d}v}{{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{y}f\left(u,v\right)\text{d}u\text{d}v}=\frac{{\int }_{-\infty }^{x}{\int }_{-\infty }^{y}f\left(u,v\right)\text{d}u\text{d}v}{{F}_{Y}\left(y\right)}$ (2.2.2)

${f}_{X|Y\le y}\left(x|Y\le y\right)=\frac{\partial {F}_{X|Y\le y}\left(x|Y\le y\right)}{\partial x}=\frac{{\int }_{-\infty }^{y}f\left(x,v\right)\text{d}v}{{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{y}f\left(u,v\right)\text{d}u\text{d}v}=\frac{{\int }_{-\infty }^{y}f\left(x,v\right)\text{d}v}{{F}_{Y}\left(y\right)}$ (2.2.3)

$\left(X,Y\right)$ 为二维离散型随机变量时，对一切使得 ${F}_{Y}\left(y\right)>0$ 的y，称

$\begin{array}{c}{F}_{X|Y\le y}\left(x|Y\le y\right)=P\left(X\le x|Y\le y\right)=\frac{P\left(X\le x,Y\le y\right)}{P\left(Y\le y\right)}\\ =\frac{\underset{{x}_{i}\le x}{\sum }\underset{{y}_{j}\le y}{\sum }P\left(X={x}_{i},Y={y}_{j}\right)}{\underset{{x}_{i}}{\sum }\underset{{y}_{j}\le y}{\sum }P\left(X={x}_{i},Y={y}_{j}\right)}=\frac{\underset{{x}_{i}\le x}{\sum }\underset{{y}_{j}\le y}{\sum }P\left(X={x}_{i},Y={y}_{j}\right)}{{F}_{Y}\left(y\right)}\end{array}$ (2.2.4)

${p}_{{x}_{i}|Y\le y}=P\left(X={x}_{i}|Y\le y\right)=\frac{P\left(X={x}_{i},Y\le y\right)}{P\left(Y\le y\right)}=\frac{\underset{{y}_{j}\le y}{\sum }P\left(X={x}_{i},Y={y}_{j}\right)}{\underset{{x}_{i}}{\sum }\underset{{y}_{j}\le y}{\sum }P\left(X={x}_{i},Y={y}_{j}\right)}=\frac{\underset{{y}_{j}\le y}{\sum }P\left(X={x}_{i},Y={y}_{j}\right)}{{F}_{Y}\left(y\right)}$ (2.2.5)

1) 当 $a<0 时，

$F\left(x|a

2) 当 $0 时，

$F\left(x|a

3) 当 $a<1 时，

$F\left(x|aa\right)}{P\left(X>a\right)}=\left\{\begin{array}{l}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\le a\\ \frac{{x}^{2}-{a}^{2}}{1-{a}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}a

$F\left(x|X+Y=n\right)=\underset{k\le x}{\sum }{C}_{n}^{k}{p}^{k}{\left(1-p\right)}^{k-1}$

3. 条件分布下的数字特征

3.1. 条件期望

$E\left(X|Y=y\right)=\underset{i}{\sum }{x}_{i}P\left(X={x}_{i}|Y=y\right)$ (3.1.1)

$E\left(X|Y=y\right)={\int }_{-\infty }^{\infty }x{f}_{X|Y}\left(x|y\right)\text{d}x$ (3.1.2)

$E\left(\left({a}_{0}+\underset{i=1}{\overset{n}{\sum }}{a}_{i}{X}_{i}\right)|Y\right)={a}_{0}+\underset{i=1}{\overset{n}{\sum }}{a}_{i}E\left({X}_{i}|Y\right)$${a}_{i}\in R$$i=1,\cdots ,n$ (3.1.3)

$E\left(h\left(Y\right)|Y\right)=h\left(Y\right)$ (3.1.4)

$E\left(g\left(X\right)h\left(Y\right)|Y\right)=h\left(Y\right)E\left(g\left(X\right)|Y\right)$ (3.1.5)

$E\left(X\right)=E\left(E\left(X|Y\right)\right)$ (3.1.7)

$\begin{array}{c}E\left(X\right)=E\left(X|Y\le {y}_{1}\right)\cdot {F}_{Y}\left({y}_{1}\right)+E\left(X|{y}_{1}{y}_{n}\right)\cdot \left(1-{F}_{Y}\left({y}_{n}\right)\right)\end{array}$ (3.1.8)

$EX=E\left(X|Y\le y\right)\cdot {F}_{Y}\left(y\right)+E\left(X|Y>y\right)\cdot \left(1-{F}_{Y}\left(y\right)\right)$ (3.1.9)

$\left(X,Y\right)$ 为二维连续随机变量时，

$\begin{array}{c}E\left(X|Y\le y\right)\cdot {F}_{Y}\left(y\right)={\int }_{-\infty }^{\infty }x{f}_{X|Y\le y}\left(x|Y\le y\right)\text{d}x\cdot {F}_{Y}\left(y\right)={\int }_{-\infty }^{\infty }x\frac{{\int }_{-\infty }^{y}f\left(x,v\right)\text{d}v}{{F}_{Y}\left(y\right)}\text{d}x\cdot {F}_{Y}\left(y\right)\\ ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{y}f\left(x,v\right)\text{d}v\text{d}x\end{array}$

$\begin{array}{c}E\left(X|Y>y\right)\cdot \left(1-{F}_{Y}\left(y\right)\right)={\int }_{-\infty }^{\infty }x{f}_{X|Y>y}\left(x|Y>y\right)\text{d}x\cdot \left(1-{F}_{Y}\left(y\right)\right)={\int }_{-\infty }^{\infty }x\frac{{\int }_{y}^{\infty }f\left(x,v\right)\text{d}v}{1-{F}_{Y}\left(y\right)}\text{d}x\cdot \left(1-{F}_{Y}\left(y\right)\right)\\ ={\int }_{-\infty }^{\infty }{\int }_{y}^{\infty }f\left(x,v\right)\text{d}v\text{d}x\end{array}$

$\begin{array}{c}E\left(X|Y\le y\right)\cdot {F}_{Y}\left(y\right)+E\left(X|Y>y\right)\cdot \left(1-{F}_{Y}\left(y\right)\right)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{y}f\left(x,v\right)\text{d}v\text{d}x+{\int }_{-\infty }^{\infty }{\int }_{y}^{\infty }f\left(x,v\right)\text{d}v\text{d}x\\ ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }xf\left(x,v\right)\text{d}v\text{d}x\\ =EX\end{array}$

$\begin{array}{c}E\left(g\left(X\right)\right)=E\left(g\left(X\right)|Y\le {y}_{1}\right)\cdot {F}_{Y}\left({y}_{1}\right)+E\left(g\left(X\right)|{y}_{1}{y}_{n}\right)\cdot \left(1-{F}_{Y}\left({y}_{n}\right)\right)\end{array}$ (3.1.10)

$\begin{array}{c}E\left({X}^{k}\right)=E\left({X}^{k}|Y\le {y}_{1}\right)\cdot {F}_{Y}\left({y}_{1}\right)+E\left({X}^{k}|{y}_{1}{y}_{n}\right)\cdot \left(1-{F}_{Y}\left({y}_{n}\right)\right)\end{array}$ (3.1.11)

$E\left[E\left(g\left(X\right)|Y\right)\right]=Eg\left(X\right)$ (3.1.12)

$E\left(XY\right)=E\left(YE\left(X|Y\right)\right)$ (3.1.13)

$Cov\left(Y,E\left(X|Y\right)\right)=Cov\left(X,Y\right)$ (3.1.14)

$E\left(XYZ\right)=E\left(XE\left(YE\left(Z|\left(X,Y\right)\right)|X\right)\right)$ (3.1.15)

1) $\begin{array}{c}E\left[E\left(g\left(X\right)|Y\right)\right]={\int }_{-\infty }^{\infty }\left({\int }_{-\infty }^{\infty }g\left(x\right){f}_{X|Y}\left(x|y\right)\text{d}x\right){f}_{Y}\left(y\right)\text{d}y\\ ={\int }_{-\infty }^{\infty }\left({\int }_{-\infty }^{\infty }g\left(x\right)\frac{f\left(x,y\right)}{{f}_{Y}\left(y\right)}\text{d}x\right){f}_{Y}\left(y\right)\text{d}y\\ ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }g\left(x\right)f\left(x,y\right)\text{d}x\text{d}y\\ =Eg\left(X\right)\end{array}$

2) $\begin{array}{c}E\left(XY\right)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }xyf\left(x,y\right)\text{d}x\text{d}y\\ ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }xy{f}_{X|Y}\left(x|y\right){f}_{Y}\left(y\right)\text{d}x\text{d}y\\ ={\int }_{-\infty }^{\infty }y\left({\int }_{-\infty }^{\infty }x{f}_{X|Y}\left(x|y\right)\text{d}x\right){f}_{Y}\left(y\right)\text{d}y\\ ={\int }_{-\infty }^{\infty }yE\left(X|Y=y\right){f}_{Y}\left(y\right)\text{d}y\\ =E\left(YE\left(X|Y\right)\right)\end{array}$

3) $\begin{array}{c}Cov\left(Y,E\left(X|Y\right)\right)=E\left(YE\left(X|Y\right)\right)-E\left(Y\right)E\left(E\left(X|Y\right)\right)\\ =E\left(XY\right)-E\left(Y\right)E\left(X\right)=Cov\left(X,Y\right)\end{array}$

4) 假设给定 $X=x$ 时， $\left(Y,Z\right)$ 的条件密度函数为 ${f}_{\left(Y,Z\right)|X}\left(\left(y,z\right)|x\right)$ ，给定 $X=x$$Y=y$ 时，Z的条件密度函数为 ${f}_{Z|\left(X,Y\right)}\left(z|\left(x,y\right)\right)$ ，且 ${f}_{\left(Y,Z\right)|X}\left(\left(y,z\right)|x\right)={f}_{Z|\left(X,Y\right)}\left(z|\left(x,y\right)\right){f}_{Y|X}\left(y|x\right)$ 成立。

$X=x$ 时，

$\begin{array}{c}E\left(YZ|X=x\right)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }yz{f}_{\left(Y,Z\right)|X}\left(\left(y,z\right)|x\right)\text{d}y\text{d}z={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }yz{f}_{Z|\left(X,Y\right)}\left(z|\left(x,y\right)\right){f}_{Y|X}\left(y|x\right)\text{d}y\text{d}z\\ ={\int }_{-\infty }^{\infty }y\left({\int }_{-\infty }^{\infty }z{f}_{Z|\left(X,Y\right)}\left(z|\left(x,y\right)\right)\text{d}z\right){f}_{Y|X}\left(y|x\right)\text{d}y={\int }_{-\infty }^{\infty }yE\left(Z|\left(X=x,Y=y\right)\right){f}_{Y|X}\left(y|x\right)\text{d}y\\ =E\left(YE\left(Z|\left(X=x,Y\right)\right)|X=x\right)\end{array}$

$E\left(YZ|X\right)=E\left(YE\left(Z|\left(X,Y\right)\right)|X\right)$

$E{\left(X-E\left(X|Y\right)\right)}^{2}\le E{\left(X-g\left(Y\right)\right)}^{2}$ (3.1.16)

3.2. 条件方差

$D\left(X|Y\right)=E\left[{\left(X-E\left(X|Y\right)\right)}^{2}|Y\right]$ (3.2.1)

$D\left(X|Y\right)=E\left({X}^{2}|Y\right)-{\left(E\left(X|Y\right)\right)}^{2}$ (3.2.2)

$D\left(c|Y\right)=0$$\forall c\in R$ (3.2.3)

$D\left(\left(aX+b\right)|Y\right)={a}^{2}D\left(X|Y\right)$$\forall a,b\in R$ (3.2.4)

$D\left(X\right)=ED\left(X|Y\right)+DE\left(X|Y\right)$ (3.2.5)

1) $\begin{array}{c}D\left(X|Y\right)=E\left[{\left(X-E\left(X|Y\right)\right)}^{2}|Y\right]\\ =E\left({X}^{2}|Y\right)+{\left(E\left(X|Y\right)\right)}^{2}-2E\left(XE\left(X|Y\right)|Y\right)\\ =E\left({X}^{2}|Y\right)+{\left(E\left(X|Y\right)\right)}^{2}-2{\left(E\left(X|Y\right)\right)}^{2}\\ =E\left({X}^{2}|Y\right)-{\left(E\left(X|Y\right)\right)}^{2}\end{array}$

2) $D\left(c|Y\right)=E\left[{\left(c-E\left(c|Y\right)\right)}^{2}|Y\right]=E\left[{\left(c-c\right)}^{2}|Y\right]=0$

3) $\begin{array}{c}D\left(\left(aX+b\right)|Y\right)=E\left[{\left(aX+b-E\left(\left(aX+b\right)|Y\right)\right)}^{2}|Y\right]\\ =E\left[{\left(aX-aE\left(X|Y\right)\right)}^{2}|Y\right]\\ ={a}^{2}E\left[{\left(X-E\left(X|Y\right)\right)}^{2}|Y\right]\\ ={a}^{2}D\left(X|Y\right)\end{array}$

4) $\begin{array}{l}D\left(X\right)=E{\left(X-EX\right)}^{2}\\ =E{\left(X-E\left(X|Y\right)+E\left(X|Y\right)-EX\right)}^{2}\\ =E{\left(X-E\left(X|Y\right)\right)}^{2}+E{\left(E\left(X|Y\right)-EX\right)}^{2}+2E\left[\left(X-E\left(X|Y\right)\right)\left(E\left(X|Y\right)-EX\right)\right]\\ =E{X}^{2}+{\left(E\left(X|Y\right)\right)}^{2}-2E\left(XE\left(X|Y\right)\right)+E{\left(E\left(X|Y\right)-E\left(E\left(X|Y\right)\right)\right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+2E\left[XE\left(X|Y\right)-XEX-{\left(E\left(X|Y\right)\right)}^{2}+EXE\left(X|Y\right)\right]\end{array}$

$\begin{array}{l}=E{X}^{2}+{\left(E\left(X|Y\right)\right)}^{2}-2E\left(XE\left(X|Y\right)\right)+DE\left(X|Y\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+2E\left(XE\left(X|Y\right)\right)-2{\left(EX\right)}^{2}-2{\left(E\left(X|Y\right)\right)}^{2}+2{\left(EX\right)}^{2}\\ =E{X}^{2}-{\left(E\left(X|Y\right)\right)}^{2}+DE\left(X|Y\right)\\ =E\left(E\left({X}^{2}|Y\right)\right)-E{\left(E\left(X|Y\right)\right)}^{2}+DE\left(X|Y\right)\\ =ED\left(X|Y\right)+DE\left(X|Y\right)\end{array}$

4. 展望

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[2] 何书元. 概率论[M]. 北京: 北京大学出版社, 2006: 177-183.

[3] 叶仁玉, 桂春燕. 不同类型随机变量之间的条件分布及其应用[J]. 安庆师范学院学报(自然科学版), 2013, 19(2): 17-19.

[4] 魏艳华, 李艳颖, 王丙参. 条件期望的性质及求法[J]. 牡丹江大学学报, 2009, 18(9): 116-117.

[5] 江五元, 丁卫平. 二维随机变量条件分布函数教学的思考[J]. 数学理论与应用, 2013, 33(4): 126-128.

[6] 吕文华, 韩慧霞. 关于条件分布及其应用的教学研究[J]. 滁州学院学报, 2014, 16(2): 126-127.

[7] 胡晓华, 虞敏. 边缘分布条件分布的几何意义及推广[J]. 大学数学, 2014, 30(3): 81-86.

[8] 宁健荣, 周玲. 条件分布计算的几个问题研究[J]. 大学数学, 2016, 32(5): 61-66.

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