(a) Let X denote the time in hours from the start of the interval until the first visit. Assume that X has an exponential distribution with parameter = 10. (a) Determine the probability that the time until the next log-on exceeds 0.25 hours. [2] (b) Determine the length of time t such that the probability that the time until the next log-on exceeds t is 0.90, that is, find the value of t such that P(X > t) = 0.90. [2] (b) The lifetime (in hours) of a certain kind of light bulb is an exponential random variable with mean value 200. Determine the probability that exactly 2 out of the 5 such bulbs will have to be replaced within the first 150 hours of operation (assume independence). [5] (c) Let X be an exponential random variable with parameter > 0. (1) Show that E(X") = E(X^-4), for n = 1,2,3,.... [4] (ii) Using your results from Part(a), find Var(X). [2]