作者LITTLEN (有没有那麽虽阿~~~)
站内Statistics
标题Re: [问题] 顺序统计问题
时间Sat Dec 11 08:21:17 2010
※ 引述《Hshs (免钱没人爱@@)》之铭言：
: Let X1 X2 X3 are the order statistics of the iid random variables X1 X2 and
: X3 with common exponential distribution with mean 1/λ. Show that
: Y1 = X3 - X2 and Y2 = X2 are independent
: 请问一下该怎麽解呢? 谢谢.
: 似乎是考古题,好像看过
f(x1,x2,x3)= 6 λ^3 exp (-λ(x1+x2+x3)) 0 < x1 < x2 < x3 < inf
f(x2,x3) = ∫6 λ^3 exp (-λ(x1+x2+x3)) dx1 (积分范围0< x1 < x2)
f(x2,x3) = 6 λ^2 exp (-λx3) [exp (-λx2) - exp (- 2λx2) ] 0< x2 < x3 < inf
Let Y1= X3 - X2 and Y2 = X2 ==> X3 = Y1 + Y2 (0 < Y2 < Y1+Y2 < inf)
By Jacobian
f(Y1,Y2)= 6 λ^2 exp (-λ(Y1+Y2)) [exp (-λY2) - exp (- 2λY2) ] * 1
= 6 λ^2 exp (-λY1) [exp (-2λY2) - exp (- 3λY2) ]
积分范围可以拆成 0< Y1 < inf 跟 0 < Y2 < inf
f(Y1,Y2) 可以拆成两个 f(Y1) * f(Y2) 所以independent
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不知道有没有误...
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