Question I (The random Telegraph Signal Process) Let {N(t).t > 0} denote a Poisson process with rate 2, and let Xbe inde

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Question I The Random Telegraph Signal Process Let N T T 0 Denote A Poisson Process With Rate 2 And Let Xbe Inde 1 (31.22 KiB) Viewed 21 times

Question I The Random Telegraph Signal Process Let N T T 0 Denote A Poisson Process With Rate 2 And Let Xbe Inde 2 (24.41 KiB) Viewed 21 times

Question I The Random Telegraph Signal Process Let N T T 0 Denote A Poisson Process With Rate 2 And Let Xbe Inde 3 (33.12 KiB) Viewed 21 times

Question I The Random Telegraph Signal Process Let N T T 0 Denote A Poisson Process With Rate 2 And Let Xbe Inde 4 (18.06 KiB) Viewed 21 times

Question I (The random Telegraph Signal Process) Let {N(t).t > 0} denote a Poisson process with rate 2, and let Xbe independent of this process and be such that P{Xo = 1) = P{Xo = -1) = 2 Defining X(t) = X. (-1)N) then {X(t),t > 0) is called a random telegraph signal process. To see that it is stationary, note first that starting at any time t, no matter what the value of (!), as X, is equally likely to be either plus or minus 1, it follows that X(t) is equally likely to be either plus or minus 1. Hence, because the continuation of a Poisson process beyond any time remains a Poisson process, it follows that {X(t),t 0} is a stationary process. 1. Calculate E[Xo) and Var[Xol. 2. Compute E[X(O) the mean function of the random telegraph signal. 3. Compute Cov[X (), X(t + s) the covariance function of the random telegraph signal.

Question 3 (Transforming a process into a Markov Chain) Suppose that whether it is aim today depends on previous edition through the last two days Specifically, suppose that it has rained for the past two days, then it will incrow with probability os fiind day heyetmay, that will now with probability 0.5 red yesterday but not day, then it will ratow with probability did it has no ruined in the past two days, their will rainbow with probability If we let the state at time depend only whether or not it is raining at times, then the preceding models 02 Markere che lowever, we can transform this delitos Marlow chain by saying that the state my time is determined by the weather conditions during out that day and the previous day. In other wonde we can say that the perces is in state riddy and yesterday state trained today but tyndy state 2 iitried yesterday but not today, state 3u did not in either yesterday or today The preceding would then represent for the Marlow hain having a retice probability matrix 0.6 0040 P. 050 0.50 0 0.40 0.6 0 0.2008 the data of the chain 1. This chain is te eine which wird which 4 Pind the price G hatine Mendoy, what is the probability that it will Thunday Geen the trained on Monday Tuesday, what is the probability that will in on Thursday

Question 4: (People immigration) Suppose that people immigrate into a territory at a Poisson process with rate 1 = 1 per day. 1. Write the definition of a standard Poisson process. 2. What is the expected time until the tenth immigrant arrives? 3. Find the probability that the elapsed time between the tenth and the eleventh arrival exceeds 2 days. Question 5: (Bicycle race) In a bicycle race between two competitors, let Y(t) denote the amount of time (in seconds) by which the racer that started in the inside position is ahead when 100 percent of the race has been completed, and suppose that {Y(t), 2 0} can be effectively modeled as a standard Brownian motion process. 1. Write the definition of a standard Brownian motion. 2. What is the distribution of y(t) - Y(s)? 3. If the inside racer is leading by 1 seconds at the midpoint of the race, calculate P(Y(1) >01Y) = 1) the probability that she is the winner?

Question 6: (Geometric Brownian Motion) Let W = {W(t), t2 0} be a standard Brownian motion. Define the Geometric Brownian Motion by X(t) = exp{W(t)}, for all t 2 0. 1. Find E[X(t)) and Var(X(t)), for all t > 0. 2. Determine E(exp{W(t) - W(s)}), for all 0 SS<t. 3. Determine Cov(x(s), X(t)), for all 0 Ss <t.