MA2102

Probability and Statistics

Lecture-21

Conditional Expectation:

Let $(X,Y)$ be a bivariate random variable then conditional expectation of $g(X,Y)$ given $Y=y$ is denoted by $E(g(X,Y)/Y=y)$ and defined as (provided sum/integral converges absolutely)

$E(g(X,Y)/Y=y)=\left\{\begin{matrix} \underset{x\in S_X}{\sum}g(x,y)p_{X/Y}(x/y)& \text{if } (X,Y) \text{ is of discrete type and } p_Y(y)>0 \\ \int_{-\infty}^{\infty}g(x,y)f_{X/Y}(x/y)dx& \text{if } (X,Y) \text{ is of continuous type and } f_Y(y)>0 \end{matrix}\right.$

similarly you can define $E(g(X,Y)/X=x)$