A computer system carries out tasks submitted by two users. Time is divided into slots. A slot can be idle, with probability pI = 1/6, and busy with probability pB = 5/6. During a busy slot, there is probability p1|B = 2/5 (respectively, p2|B = 3/5) that a task from user 1 (respectively, 2) is executed. We assume that events related to different slots are independent.
(a) Find the probability that a task from user 1 is executed for the first time during the 4th slot.
(b) Given that exactly 5 out of the first 10 slots were idle, find the probability that the 6th idle slot is 12.
(c) Find the expected number of slots up to and including the 5th task from user 1.
(d) Find the expected number of busy slots up to and including the 5th task from user 1.
(e) Find the PMF, mean, and variance of the number of tasks from user 2 until the time of the 5th task from user 1.