The joint frequency distribution of two discrete random variables, and , is given in the following table
a) Can anybody help me how to calculate the conditional expectation of for a given value of : (|=). For example, similar to the formula in we have: (|=1)=Σ⋅(=|=1). Find (|=1) and
b) Calculate (|=3)
It is very easy. Fixed a value of $X=x$ in the above table you will see which are the values of $Y$ (in this case for any value of $X$ you have $Y=1,2,3,4$ but it could be not necessary so, in general) and all you have to do is to calulate the conditional probability with the usual formula
Example, fix $X=3$ and you get
$$(Y|X=3)=\begin{cases} 2/31, & \text{if $Y=1$} \\ 5/31, & \text{if $Y=2$} \\ 20/31, & \text{if $Y=3$} \\ 4/31, & \text{if $Y=4$} \end{cases}$$
The expectation follows as a consequence being
$$E(Y|X=x)=\Sigma_y yp(y|x)$$