Discrete-time stochastic processes

N tt is the time average renewal rate over the

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Unformatted text preview: in a given town arrive at the place of voting according to a Poisson process of rate ∏ = 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 − 1st . For example, in the sequence of votes AAB AAB B , 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. Exercise 2.23. Let {N1 (t); t ≥ 0} be a Poisson counting process of rate ∏. Assume that the arrivals from this process are switched on and off by arrivals from a second independent Poisson process {N2 (t); t ≥ 0} of rate ∞ . Let {NA (t); t≥0} be the switched process; that is NA (t) includes the arrivals from {N1 (t); t ≥ 0} during periods when N2 (t) is even and excludes the arrivals from {N1 (t); t ≥ 0} while N2 (t) is odd. 2.7. EXERCISES rate ∏ rate ∞ 89 ❆ ✁ ❆ ✁ ✛ On ✲ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ✛ On ✲ ❆ ✁ ❆❆ ✁✁ ❆ ✁ ❆ ✁ ✛ ❆ ✁ ❆ ✁ N2 (t) ✲ On ❆ ✁ N1 (t) ❆ ✁ NA (t) a) Find the PMF for the number of arrivals of the first process, {N1 (t); t ≥ 0}, during the nth period when the switch is on. b) Given that the first arrival for the second process occurs at epoch τ , find the conditional PMF for the number of arrivals of the first process up to τ . c) Given that the number of arrivals of the first process, up to the first arrival for the second process, is n, find the density for the epoch of the first arrival from the second process. d) Find the density of the interarrival time for {NA (t); t ≥ 0}. Note: This part is quite messy and is done most easily via Laplace transforms. Exercise 2.24. Let us model the chess tournament between Fisher and Spassky as a stochastic process. Let Xi , for i ≥ 1, be the duration of the ith game and assume that {Xi ; i≥1} is a set of IID exponentially distributed rv’s each with density f (x) = ∏e−∏x . Suppose that each game (independently of all other games, and independently of the length of the games) is won by Fisher with probability p, by Spassky with probability q , and is a draw with probability 1 − p − q . The first player to win n games is defined to be the winner, but we consider the match up to the point of winning as being embedded in an unending sequence of games. a) Find the distribution of time, from the beginning of the match, until the completion of the first game that is won (i.e., that is not a draw). Characterize the process of the number {N (t); t ≥ 0} of games won up to and including time t. Characterize the process of the number {NF (t); t ≥ 0} of games won by Fisher and the number {NS (t); t ≥ 0} won by Spassky. b) For the remainder of the problem, assume that the probability of a draw is zero; i.e., that p + q = 1. How many of the first 2n − 1 games must be won by Fisher in order to win the match? c) What is the probability that Fisher wins the match? Your answer should not involve any integrals. Hint: consider the unending sequence of games and use part b. d) Let T be the epoch at which the match is completed (i.e., either Fisher or Spassky wins). Find the distribution function of T . e) Find the probability that Fisher wins and that T lies in the interval (t, t + δ ) for arbitrarily small δ . Exercise 2.25. Using (2.43), find the conditional density of Si+1 , conditional on N (t) = n and Si = si , and use this to find the joint density of S1 , . . . , Sn conditional on N (t) = n. Verify that your answer agrees with (2.34). 90 CHAPTER 2. POISSON PROCESSES Exercise 2.26. A two-dimensional Poisson process is a process of randomly occurring special points in the plane such that (i) for any region of area A the number of special points in that region has a Poisson distribution with mean ∏A, and (ii) the number of special points in nonoverlapping regions is independent. For such a process consider an arbitrary location in the plane and let X denote its distance from its nearest special point (where distance is measured in the usual Euclidean manner). Show that a...
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