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Math Help - Variance of a Poisson random variable whose parameter is also a random variable

  1. #1
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    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.
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  2. #2
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    Re: Variance of a Poisson random variable whose parameter is also a random variable

    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)

    = cē.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]
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