MarkDown Letex 编码 之 随机过程及应用（二） - E[X|Y] = E[Y E[X|Y]]证明

**E[X|Y] = E[Y E[X|Y]]证明：**            ||   **1.Just use the following: For an integrable random variable $Z$, 对于一个可积随机变量$Z$，$$E[Z] =E[\ E[Z|F]\ ]$$ for any $σ-algebra \ F$ . If $Y$ is $F-measurable,$                    then $$E[XY∣F]=YE[X∣F]$$.           ||   **2.Basically                                                     $$ E[XY]=E[\ E[XY|Y]\ ]=E[\ YE[X|Y]\ ]\ . $$* * The first step is the iterated rule of conditional expectation. * * 第一步是条件期望的迭代规则。* * * For the second, use the fact that given Y, Y is like a constant.* * 第二,使用假设 给出Y , Y是一个常数。* * However if you are looking for the usage of rigorous definition of conditional expectation, the solution by Davide Giraudo is the one to go for.* * 但如果你在寻找条件期望严格定义的使用,还请参考Davide Giraudo提出的方法。