※ 引述《mynameisgod (godblessyou)》之铭言： : 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 α与λ弄颠倒 αt λ α = e (-------------) λ-exp(t)+1 exp(t)λ α = (-------------) λ-exp(t)+1 exp(t)λ/(1+λ) α = (-----------------) 1-exp(t) / (1+λ) 令p=λ/(1+λ) p exp(t) α = (-----------------) 1-(1-p)exp(t) 负二项 mgf : ∞ 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 -- 好累！ --

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