1. (a) Let X be exponential random variable with parameter A, i.e. its density is f(x) = de-X, 0 1) and E(X|1 e-> for > > 0. Find E[max { X, Y, Z}]. (d) Let Z follow the exponential distribution with parameter # > 0, and given Z, X and Y are independent Poisson random variables with parameter Z. Find the conditional distribution of X given that X + Y =n where n 2 1 is a given integer. (e) Let X and Y be independent random variables distributed as exponential with parameters A and #, respectively (i.e. a probability density function (pdf) of X is fx(x) = de-> for r > 0, and a pdf of Y is fy(y) = perky for y > 0). Let I, independent of X and Y, be a Bernoulli random variable with success probability P(I = 1) = >+ / Define W =X-Y and Z = X if I = 1, -Y if I = 0. Show, by using moment generating functions (e.g. Mx (t) = Elet*] = ] for # p. Let c > 0 and 6 = A - / > 0. (i) Show that the conditional density function of X, given that X +Y = c, is fxx+y(x|c) = 1 - e-de 0