Question

Each day, starting at 5pm, Alice starts to wait for Bob’s call. It is known that each day, the waiting time for Bob’s call is an Exponential random variable with expectation 1/2 hour, and that the times of his calls are independent from day to day. Alice’s day is unhappy if she has to wait for more than log 10 hours for Bob’s call. (As log 10 hours is about 2.3 hours, Alice is not being unreasonable.)

(a) Compute the probability that tomorrow is Alice’s unhappy day.

(b) Let X be the number of Alice’s unhappy days in the next 200 days. Identify the distribution of X (as one of the famous probability mass functions.) Determine EX.

(c) Using an appropriate approximation, approximate P(X ≥ 3).

Answer 1