Thread: Variance of a Poisson random variable whose parameter is also a random variable

Hello,

be k a Poisson random variable of parameter lambda.
The parameter lambda be lambda=c j where c is a constant and j is a Poisson random variable of parameter gamma.

Now:
1- var j= gamma
2- var lambda = c^2 gamma
3- var k = lambda is not true, what is it?

Many thanks.

Var(K) = Var(E(K|λ)) + E(Var(K|λ))

E(K|λ) = λ ...k a Poisson random variable

Var(K|λ)= λ ...k a Poisson random variable

Var(K) = Var(λ) + E(λ)

= c².Var(J) + cE(J) .... λ =c*J

J is a Poisson random variable of parameter gamma,γ

= c.[c.Var(J) + E(J)]

= cγ[c+1]