推 hata506:太厲害了 謝謝指點 ^^ 10/19 21:33
let { Xn , n = 1, 2, ..... } be a set of iid Exp(λ) random variables,
and let N be an independent G(p) random variable.
Show that Z = X_1 + X_2 + ... + X_N
is an Exp(λp) random variable.
拜託各位大大了
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作者: mangogogo (ka) 看板: Statistics
標題: Re: [問題] 證明分配
時間: Wed Oct 19 17:01:34 2005
※ 引述《hata506 (真夏的聖誕節)》之銘言：
: let { Xn , n = 1, 2, ..... } be a set of iid Exp(λ) random variables,
: and let N be an independent G(p) random variable.
: Show that Z = X_1 + X_2 + ... + X_N
: is an Exp(λp) random variable.
: 拜託各位大大了
Z=ΣX_i|N~Γ(N,λ)
N~G(p)
λ ∞ λ
M(t)=E[exp{-Zt}]=EE[exp{-Zt}|N]=E[(------)^N]=Σ(------)^n*(p(1-p)^n-1)
λ-t n=1 λ-t
p λ(1-p) p v
=----Σ[------]^n=-----*------ λ(1-p)
1-p λ-t 1-p 1-v (令v=-------)
λ-t
λp
=--------
λp-t
由動差生成函數的唯一性 可以知道Z~Exp(λp)
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