Let X and Y have the joint pmf (a) Display the joint pmf

QUESTION:

Let \(X\) and \(Y\) have the joint pmf

\(f(x, y)=\frac{x+y}{32}, \quad x=1,2, \quad y=1,2,3,4\) .

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

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

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

(d) Find \(P(1 \leq Y \leq 3 \mid X=1), P(Y \leq 2 \mid X=2)\), and \(P(X=2 \mid \bar{Y}=3)\).

(e) Find \(E(Y \mid X=1)\) and \(\operatorname{Var}(Y \mid X=1)\).