''An auditor is checking tax returns for fraud. They need to find Y returns with irregularities to review in detail and Y is known to be Poisson with mean 10. The probability that any particular return has an irregularity is 0.05 (independent).

Let X=total number of returns the auditor checks

a)What is the conditional distribution of X|Y=y?

b)Find E(X|Y=y) and Var(X|Y=y)

c)Find E(X|Y) and Var(X|Y)

d)Find E(X)

e)Find Var(X)

f)For each return that does NOT have an irregularity, the cost in lost productivity has a gamma distribution with = 3 and = 4. Calculate the mean and variance of the Total Cost.