出处:mathematical statistics and data analysis John A. RICE Page.172 #92 Suppose that Θ is a random variable that follows a gamma distribution with parameter λ and α,where α is an integer, and suppose that, conditional on Θ,Χ follows a Poisson distribution with parameter Θ. Find the unconditional distribution of α+Χ. (Hind:Find the mgf by using iterated conditional expectations.) my solution: t(α+x) αt tx αt tx αt θ(exp(t)-1)) Mα+X(t) = E(e ) = e E(e ) = e E(E(e |θ)) = e E(e ) αt t αt α λ = e Mθ(e - 1) = e (-------------) 然後我就看不出来他是哪种distribution α-exp(t)+1 ∞ 1 α α-1 -λx 这本书的Gamma:X~Γ(α,λ) f(x)=∫------λ x e dx (和Hogg不同) oΓ(α) 因为题目是写with parameter λ and α 所以我代的 distribution 为Γ(λ,α) Γ(y+λ-α) λ -y-λ+α P.S.我一开始硬解pdf f_Y(y)=----------------α (1+α) ,Y=X+α (y-α)!Γ(λ) 还是看不出来是什麽玩意儿 所以上来请各位神手帮帮忙 --

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