Exercise 5.41 Compound distributions. If Y ~ Poisson(A) then E(Y)= \ and Var(Y) d. The Poisson distribution has only a single parameter so we cannot shift and scale the = distribution by different...

Extracted text: Exercise 5.41 Compound distributions. If Y ~ Poisson(A) then E(Y)= \ and Var(Y) d. The Poisson distribution has only a single parameter so we cannot shift and scale the = distribution by different amounts, which limits its usefulness in certain practical applications. This problem can be addressed by allowing the parameter itself to be a random variable. 1. Let Y ~ Poisson(X) where X ~ Exponential (0) and 0 > 0 is a fixed scale parameter. Use the laws of total expectation and total variance and the fact that E(X) = 0 and Var(X) = 0² to show that E(Y) = 0 and Var(Y) = 0(1+0).