Refer to Exercise 9.

a. Find the conditional probability mass function \(p_{Y \mid X}(y \mid 4)\).

b. Find the conditional probability mass function \(p_{X \mid Y}(x \mid 3)\).

c. Find the conditional expectation \(E(Y \mid X=4)\).

d. Find the conditional expectation \(E(X \mid Y=3)\).

Data From Exercise 9:

The General Social Survey asked a sample of adults how many siblings (brothers and sisters) they had \((X)\) and also how many children they had \((Y)\). We show results for those who had no more than 4 children and no more than 4 siblings. Assume that the joint probability mass function is given in the following contingency table.

X 01234 2 O 0.03 0.09 0.09 0.06 0.04 y 1 0.01 0.02 0.05 0.08 0.05 0.04 0.03 2 0.07 0.07 0.04 3 0.01 0.03 0.04 0.04 0.03 4 0.01 0.01 0.02 0.02 0.02