Suppose 1,2, ... , are independent exponential random variables with rates 1, 2, ... , .

Suppose X1,X2, , X n are independent exponential random variables with rates 11,12, , lln. That is, the probability function of Xi is, Aie'ix, x > 0 0, otherwise Mac.) ={ a) Let Y: Inin{X2, X3}, i) (3 marks) Find the probability density function of Y. ii) (3 marks) Find the moment generating mction of Y. iii) (4 marks) Use My(t) to solve for the expected value and variance of Y. b) (5 marks) Suppose n =105, and that A, = 0.25 for all i=1, 2, ...,105. Then X1,X2, ,X105 are independent and identically distributed exponential random variables. Let I? = i 213? X i. Using the central limit theorem state the distribution of X and compute the probability that P(3.8