The voters in a given town arrive at the place of voting according to a Poisson process of rate λ = 100 voters per hour. The voters independently vote for candidate A and candidate B each with probability 1/2. Assume that the voting starts at time 0 and continues indefinitely.
(a) Conditional on 1000 voters arriving during the first 10 hours of voting, find the probability that candidate A receives n of those votes.
(b) Again conditional on 1000 voters during the first 10 hours, find the probability that candidate A receives n votes in the first 4 hours of voting.
(c) Let T be the epoch of the arrival of the first voter voting for candidate A. Find the density of T.
(d) Find the PMF of the number of voters for candidate B who arrive before the first voter for A.
(e) Define the nth voter as a reversal if the nth voter votes for a different candidate than the (n-1)th. For example, in the sequence of votes AABAABB, the third, fourth, and sixth voters are reversals; the third and sixth are A to B reversals and the fourth is a B to A reversal. Let N(t) be the number of reversals up to time t (t in hours). Is {N(t); t > 0} a Poisson process? Explain.
(f) Find the expected time (in hours) between reversals.
(g) Find the probability density of the time between reversals.
(h) Find the density of the time from one A to B reversal to the next A to B reversal.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.