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3. (20 points) Suppose X ~ Exp(1) and Y = -In(X) (a) Find h(y) = P(Y > y) (b) Find the probability density function of Y. (c) Find the probability density function of W = -In(T), where T = h(Y) dy (Hint : aFy (y) = fy (y) (d) Let X1, X2, . . ., Xk bei.i.d. Exp(1), and let Mk = max(X1, . .. , Xk) (Maximum of X1, ..., Xk). Find the probability density function of Mk. (Hint: P(min(X1, X2, X3) > K) = P(X1 > k, X2 > k, X3 > k), how about max ?) (e) Show that as k - oo, the CDF of Z = Mk - In(k) is the CDF of Y. Hint: Note that as k - oo, (1 + 2)k = ed. (f) Suppose there is a variable 'A' whose probability density function is exactly same as the probability density function of W and A is independent of X, Y and W. Find the probability density function of B = Ate-Y