Problem 4: Alice, Bob and Carmen are playing a game of hot potato, with the following rules. Alice starts with the potato at time t = 0, and at t = 1 passes it to Bob. At t = 2 Bob passes it to Carmen, then at t = 3 Carmen passes it back to Alice, and the cycle continues. Meanwhile, music is playing that will stop at a random time T , and whoever has the potato at time T will lose. (Note that T0.)

Answer the following:

(a) Suppose that T is distributed Exponential(). Conditional on T 3, what is the probability that Alice loses?

(b) Again supposing that T is distributed Exponential(), what is the probability that Alice loses? (Without conditioning on T3)

(c) Suppose that a random value X is first drawn uniformly from the interval [0, 3], and then T is randomly drawn uniformly from the interval [0, X]. Find the following:

• The joint pdf of (X, T)
• The marginal pdf of T
• The hazard rate of T
• The probability that Alice loses

Be sure to give the bounds on the values of the variables in your answers for the pdfs and hazard rate.