[機統] F(X)的order statistics distribution

[yueayase (scrya)]

Let Y1 < Y2 < ... < Y10 be the order statistics of random sample from

a continuous type distribution with cdf F(x). Find the joint distribution

of V1 = F(Y4)-F(Y2), V2 = F(Y10) - F(Y6).


我的作法:

V1=F(Y4) - F(Y2) 和 F(Y2)的分布一樣

V2=F(Y10) - F(Y6) 和 F(Y4)的分布一樣

The joint pdf of F(Y2) and F(Y4) is

               10!                   6
f(y2, y4) = --------- y2(y4-y2)(1-y4)  , 0 < y2 < y4 < 1
             1! 1! 6!


            10!      3         4
但答案是 ------- v1v2 (1-v1-v2) , 0 < v2, v2+v4 < 1
          1!3!4!


為什麼不是



               10!                   6
f(v1, v2) = --------- v1(v2-v1)(1-v2)  , 0 < v1 < v2 < 1
             1! 1! 6!


?

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