[機統] 算期望值
拜託大家幫我看看這題
看我有沒有寫錯
一個隨機變數X 有 f_X(x)=2x/a^2 , 0<x<a , 假設係數a為未知數
然後來估計 a set of n independent sample of X 之最大值
i.e.
a*=max{X_1,X_2,X3,........,X_n}
求E[a*]
E[max(X_1,X_2,X_3,......,X_n)]= P[max(X_1,X_2,X_3,....X_n)>t]dt (從0積到a)
= {1-P[max(X_1,X_2,X_3,....,X_n)<=t]}dt= {1-P[X_1<=t]*****P[X_n<=t]}dt
又 P[X_1<=t]= (2x/a^2)dx (從0積到t)
=t^2/a^2
(從0積到a)
所以原式E[max(X_1,X_2,X_3,......,X_n)]={1-P[X_1<=t]*****P[X_n<=t]}dt
=[1-(t^2/a^2)^n]dt= a- [a^(2n+1)/(a^2n)*(2n+1)]
所以 E[a*]= a- [a^(2n+1)/(a^2n)*(2n+1)]
不曉得這樣有沒有錯
謝謝大家
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