Let W1 < W2 < < Wn be the order statistics of n independent observations from a U(0, 1)

Chapter 6, Problem 6.3-6

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QUESTION:

Let W1 < W2 < < Wn be the order statistics of n independent observations from a U(0, 1) distribution. (a) Find the pdf of W1 and that of Wn. (b) Use the results of (a) to verify that E(W1) = 1/(n+1) and E(Wn) = n/(n + 1). (c) Show that the pdf of Wr is beta.

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QUESTION:

Let W1 < W2 < < Wn be the order statistics of n independent observations from a U(0, 1) distribution. (a) Find the pdf of W1 and that of Wn. (b) Use the results of (a) to verify that E(W1) = 1/(n+1) and E(Wn) = n/(n + 1). (c) Show that the pdf of Wr is beta.

ANSWER:

Step 1 of 3

The PDF of a uniformly distributed variable over (0,1) is given as,

                                                                 

Also,

                                                                         

(a)

For ,  PDF of  is,

                                                           

For ,  PDF of  is,

                                                           

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