Chapter 4.3, Problem 1E

Let X and Y have the joint pmf $f\left(x,y\right)=\frac{x+y}{32},x=1,2,y=1,2,3,4.$

(a) Display the joint prnf and the marginal pmfs on a graph like Figure 4.3-1(a).

(b) Find $g\left(x|y\right)$ and draw a figure like Figure 4.3-1(b), depicting the conditional pmfs for $y=1,2,3,$ and 4.

(c) Find $h\left(y|x\right)$ and draw a figure like Figure 4.3-1(c), depicting the conditional pmfs for $x=1$ and 2.

(d) Find $P\left(1\le Y\le 3|X=1\right),P\left(Y\le 2|X=2\right)$, and $P\left(X=2|Y=3\right)$.

(c) Find $E\left(Y|X=1\right)$ and $Var\left(Y|X=1\right)$.