Suppose X1 and X2 are independent random variables each with the uniform distrib
ID: 3083554 • Letter: S
Question
Suppose X1 and X2 are independent random variables each with the uniform distribution on (0,1). Find E[X(1)] and E[X(2)]. Thank you.Explanation / Answer
(a) P(X > x) = P(X1 > x, ..., and Xn > x) = P(X1 > x) * ... * P(Xn > x), since the random variables are independent = (1 - x) * ... * (1 - x), since each Xi is uniform on (0, 1) = (1 - x)^n. (b) P(X > x/n) = (1 - x/n)^n, by part a. Letting L = lim(n??) (1 - x/n)^n. we have ln L = lim(n??) n ln(1 - x/n) ......= lim(n??) ln(1 - x/n)/(1/n) ......= lim(t?0+) ln(1 - xt)/t, letting t = 1/n ......= lim(t?0+) [-x/(1 - xt)]/1, by L'Hopital's Rule ......= -x. Hence, L = e^(-x), as required.
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