#todo Let $(\Omega,\mathcal{F},\mathbb{P})$ be a [[Probability Space|probability space]]. Let $X,Y$ be a non-negative [[Discrete random variables|discrete real-valued random]] variable on $(\Omega,\mathcal{F},\mathbb{P}).$ Let $A\in\mathcal{F}$ so that $\mathbb{P}(B)>0.$ $\mathbb{E}[X\mid Y]$ is a DRV defined by the [[Conditional Expectation of Discrete Real Valued Random Variable Over Event|conditional expection]] $\mathbb{E}[X\mid Y=y]$ for all $y\in D_{Y}.$ # Properties $\mathbb{E}[X\mid X]=X$ $\mathbb{E}[X\mid Y]=\mathbb{E}[X]$ if $X,Y$ are independent. $\mathbb{E}[aX+bY\mid Z]=a\mathbb{E}[X\mid Z]+b\mathbb{E}[Y\mid Z]$. By [[Expectation of Conditional Expectation of Discrete Random Variable Over Another Equals Expectation (Total Expectation Theorem)]], $\mathbb{E}[\mathbb{E}[X\mid Y]]=\mathbb{E}[X].$