[數統] converge in probability
※ [本文轉錄自 Math 看板 #1DL-N5M2 ]
作者: idphobia5566 (idphobia5566) 看板: Math
標題: [分析] converge in probability
時間: Sun Feb 13 22:08:02 2011
iid
X1,...,Xn ~ UNIF(a,b) , a<b , let X(n) be the largest order statistic.
(a)What does exp[X(n)] converge in probability? Show your work.
(b)Find the limiting distribution of exp[-n(b-X(n))/(b-a)] . Show your work.
最大順序統計量之pdf: f_x:n(x) = n(x-a)^n-1/(b-a)^n , a<x<b
想法是
let y = exp(x) , 找出y=exp[X(n)]的pdf，然後用mgf法取極限來求
但這樣不知道怎麼積分...因為式子很難看
f(y)= n{[ln(y)-a]^(n-1)}/[y*(b-a)^n] , exp(a) < y < exp(b)
M_y(t)=E[exp(ty)] 超級難積分
請問有沒有更好的方法可以做，謝謝
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