Suppose is an exponential random variable with mean 1. Conditional on = , N is

Suppose Λ is an exponential random variable with mean 1. Conditional on Λ = λ, N is a Poisson random variable with parameter λ.
(a) Find E[N|Λ] and V[N|Λ]
(b) Use the law of total expectation to find E[N].
(c) Use the law of total variance to find V[N].
(d) Find the probability mass function of N.
(e) Find E[N] and V[N] again using the pmf of N.