关于随机独立性和回归独立性的几个命题的证明

有3种情形来描述随机变量X和Y之间的关系：（i）X和Y随机独立；（ii）Y回归独立于；（iii）X和Y不相关。主要证明了：1）若X和Y随机独立，则Y回归独立于X；2）若Y回归独立于X，则X和Y不相关；3）若X和Y均为两值随机变量或（X，Y）服从二维正态分布，则三种情形等价。同时得到一个推广：设Y为两值随机变量，X是离散型随机变量，若Y回归独立于X，则X和Y随机独立。

Proofs of Some Propositions about Random Independence and Regressive Independence

There are three statements to describe the relations between two random variables X and Y:(i) X and Y are random independent;(ii) Y is regressive independent of X;(iii) X and Y are uncorrelated.This article proves: 1) Statement(i) implies statement(ii);2) statement(ii) implies statement(iii);3) Statements(i),(ii) and(iii) are equivalent if X and Y are both taking only two values or(X,Y) has bivariate normal distribution.Furthermore, a deduction is given,that is: if Y is taking only two values and regressive independent of X,then X and Y are random independent.