Suppose that the conditional distribution of N, given that Y = y, is Poisson with mean y. Further suppose that Y is a gamma random variable with parameters

(r, λ), where r is a positive integer. That is, suppose that

and

(a) Find E[N].

(b) Find Var(N).

(c) Find P(N = n)

(d) Using (c), argue that N is distributed as the total number of failures before the rth success when each trial is independently a success with probability p = λ
1+λ .

Transcribed Image Text:

P(N=nY=y)=e^ n!