[問題] 求分布

[mynameisgod (godblessyou)]

出處: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   (-------------)   然後我就看不出來他是哪種distribution
                          α-exp(t)+1

                                ∞  1    α  α-1  -λx
這本書的Gamma:X~Γ(α,λ)  f(x)=∫------λ  x     e     dx     (和Hogg不同)
                                 oΓ(α)

因為題目是寫with parameter λ and α 所以我代的 distribution 為Γ(λ,α)

                              Γ(y+λ-α)    λ       -y-λ+α
P.S.我一開始硬解pdf f_Y(y)=----------------α   (1+α)          ,Y=X+α
                              (y-α)!Γ(λ)

還是看不出來是什麼玩意兒  所以上來請各位神手幫幫忙

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◆ From: 140.114.198.109
※ 編輯: mynameisgod     來自: 140.114.198.109      (02/09 00:39)
※ 編輯: mynameisgod     來自: 140.114.198.109      (02/09 00:52)
推 jasonkeen:直接作就好啦，題目只要你求分佈，不一定會是常見分佈呀