Re: [問題] 請問卜瓦松機率分配
※ 引述《yofute (Henin讚啦)》之銘言：
: p(x)=(λ^x*e^-λ)/x!，的E(x)和變異數都是λ
: 請問要怎麼分別證明呢？
: 我用x=0,1,2.....帶入 x*p(x)化簡，結果算兩個式子就卡住了@@
: 請教各位了^^ 感謝
E[x] = Σxp(x) x = 0~∞
(λ^x)e^(-λ)
= Σx-------------
x!
λ^(x-1)
= λe^(-λ)Σ--------- x = 1~∞ , let y = x-1
(x-1)!
λ^y
= λe^(-λ)Σ---- y = 0~∞
y!
= λe^(-λ)e^λ
= λ
E[x^2] = Σ(x^2)p(x) x = 0~∞
(λ^x)e^(-λ)
= Σ(x^2)-------------
x!
x*e^(-λ)[λ^(x-1)]
= λΣ------------------- x = 1~∞ , let y = x-1
(x-1)!
(y+1)e^(-λ)(λ^y)
= λΣ------------------ y = 0~∞
y!
ye^(-λ)(λ^y) e^(-λ)(λ^y)
= λ(Σ-------------- + Σ-------------)
y! y!
= λ(λ+1)
Var(x) = E[x^2] - (E[x])^2
= λ(λ+1) - λ^2
= λ
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