Question

4. Suppose a random number generator generates 20 numbers per second, where each number is drawn uniformly from the interval 9,10), independently of all other numbers. We are interested in the event that one of the drawn numbers is very close to Usain Bolt's 100m sprint world record, that is, that this number belongs to the interval (9.575, 9.585). (a) Suppose that the random number generator runs for 10 seconds. Use the Poisson approximation to estimate the probability that it produces more than 4 numbers that fall in that interval. (b) We now let the generator run indefinitely. Use the exponential random variable to estimate the probability that it produces the first such number before Usain Bolt finishes his run, i.e. within 9.58 seconds. (c) Suppose we let the random number generator compete against Usain Bolt, as described in (b), every day for 100 days. Use the normal random variable to approximate the probability that Usain Bolt wins at least 12 of these competitions.

Answer 1