A light fixture contains five lightbulbs. The lifetime of each bulb is exponentially distributed with mean 200 hours. Whenever a bulb burns out, it is replaced. Let T be the time of the first bulb replacement. Let Xi , i = 1,..., 5, be the lifetimes of the five bulbs. Assume the lifetimes of the bulbs are independent. a. Find P(X1 > 100). b. Find P(X1 > 100 and X2 > 100 and ··· and X5 > 100). c. Explain why the event T > 100 is the same as {X1 >100 and X2 >100 and ··· and X5 >100}. d. Find P(T ≤ 100). e. Let t be any positive number. Find P(T ≤ t), which is the cumulative distribution function of T . f. Does T have an exponential distribution? g. Find the mean of T . h. If there were n lightbulbs, and the lifetime of each was exponentially distributed with parameter λ, what would be the distribution of T ?