Question

W~f(w)=λe^(-λw) (w>0) (N | W) ~ Poisson [W] i) Find P[N=n] (n=0,1,2...) ii) Find the conditional...

W~f(w)=λe^(-λw) (w>0)

(N | W) ~ Poisson [W]

i) Find P[N=n] (n=0,1,2...)

ii) Find the conditional pdf of (W | N = n)

iii) Find E[W | N]

iv) Find Var[W | N]

Math   Statistics-And-Probability   
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Homework Answers

Answer #1

The pdf of W is

The conditional pmf of N|W is

The marginal pmf of N is

. Hence

this is a Negative Binomial distribution

That is the distribution of N is

ii) The conditional pdf is obtained using the formula for conditional probability

this is a gamma distribution. Hence

iii) The expectation of is

The expectation of W|N is

iv) The variance of is

The variance of W|N is

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