Deduce that the random variable

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a) If X1...Xn are independent U(0,1) random variables and Y = ( ∏ from i=1 to n of Xi)^1/n, show that nlogY has m.g.f. (1 + z)^-n. i=1 (Hint: first find the m.g.f. of log Xi. 

(b) Suppose that Y has density [n^n/(n-1)!](-y log y)^n-1 on (0,1). Show that E(Y^nz) =(1 + z)^-n. (Hint: start by substituting t = -log y.) (Here you may use the fact that integral form 0 to infinity of (s^n-1)ds = (n-1)! for n = 1, 2...) 

 

(c) Deduce that the random variable Y in (a) has the density given in (b).

Reference no: EM13688002

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