Problem 5 We select for a race a random number of runners out of a group of 8 people, and record the time T of the winner in the race. Suppose that the running times of the individual runners in the original group of 8 are independent standard exponential random variables (i.e., Exp(1)). Suppose also that we are equally likely to select any number from 1 to 8 of the runners for the race. Find the expected winning time E(T), and the variance of T. (Hint: use the fact that if Z1, . .., Zn iid ~ Exp(1), then min(Z1, ..., Zn) ~ Exp(n).) Problem 6 Let X be a geometric random variable whose probability for success is itself