Interlude: Practice Final

Transcription

1 8 POISSON PROCESS 08 Interlude: Practice Final This practice exam covers the material from the chapters 9 through 8. Give yourself 0 minutes to solve the six problems, which you may assume have equal point score.. You select a random number X in [0,] (uniformly). Your friend then keeps selecting random numbers U,U,... in [0,] (uniformly and independently) until he gets a number larger than X/, then he stops. (a) Compute the expected number N of times your friend selects a number. (b) Compute the expected sum S of the numbers your friend selects.. You are a member of a sports team and your coach has instituted the following policy. You begin with zero warnings. After every game the coach evaluates whether you ve had a discipline problem during the game; if so, he gives you a warning. After you receive two warnings (not necessarily in consecutive games), you are suspended for the next game, and your warnings count goes back to zero. After the suspension, the rules are the same as at the beginning. You figure you will receive a warning after each game you play independently with probability p (0,). (a) Let the state of Markov chain be your warning count after a game. Write down the transition matrix and determine whether this chain is irreducible and aperiodic. Compute its invariant distribution. (b) Write down an expression for the probability that you are suspended for both games 0 and 5. Do not evaluate. (c) Let s n be the probability that you are suspended in the n th game. Compute lim n s n. 3. A random walker on the nonnegative integers starts at 0, and then at each step adds either or 3 to her position, each with probability. (a) Compute the probability that the walker is at n + 3 after making n steps. (b) Let p n be the probability that the walker ever hits n. Compute lim n p n. 4. A random walker is in one of six vertices, labeled 0,,,3,4, and 5, of the graph in the picture. At each time, she moves to a randomly chosen vertex connected to her current position by an edge. (All choices are equally likely and she never stays at the same position for two successive steps.)

2 8 POISSON PROCESS 09 (a) Compute the proportion of times the walker spends at 0, after she makes many steps. Does this proportion depend on the walker s starting vertex? (b) Compute the proportion of times the walker is at an odd state (, 3, or 5) while she previously was at even state (0,, or 4). (c) Now assume that the walker starts at 0. What is expected number of steps she will take before she is back at 0? 5. In a branching process, an individual has two descendants with probability 3 4 and no descendants with probability 4. The process starts with a single individual at generation 0. (a) Compute the expected number of individuals in generation. (b) Compute the probability that the process ever goes extinct. 6. Customers arrive at a two-server service stations, labeled and, as a Poisson process with rate λ. Assume time unit is one hour. Whenever a new customer arrives, any previous customers are immediately ejected from the system. A new arrival enters the service at station, then goes to station. (a) Assume the service time at each station is exactly hours. What proportion of entering customers will complete the service (before they are ejected)? (b) Assume the service time at each station is now exponential with expectation hours. What proportion of entering customers will now complete the service? (c) Keep the service time assumption from (b). A customer arrives, but he is now given special treatment: he will not be ejected unless at least three or more new customers arrive during his service. Compute the probability that this special customer is allowed to complete his service.

3 8 POISSON PROCESS 0 Solutions to Practice Final. You select a random number X in [0, ] (uniformly). Your friend then keeps selecting random numbers U,U,... in [0,] (uniformly and independently) until he gets a number larger than X/, then he stops. (a) Compute the expected number N of times your friend selects a number. Given X = x, N is distributed geometrically with success probability x, so and so, E[N X = x] = x, EN = 0 dx x = log( x ) 0 = log. (b) Compute the expected sum S of the numbers your friend selects. Given X = x and N = n, your friend selects n numbers uniformly in [0, x ] and one number uniformly in [ x,]. Therefore, E[S X = x,n = n] = (n ) x 4 + ( + x ) = 4 nx +, E[S X = x] = 4 x x + = x, ES = dx = log. x 0. You are a member of a sports team and your coach has instituted the following policy. You begin with zero warnings. After every game the coach evaluates whether you ve had a discipline problem during the game; if so, he gives you a warning. After you receive two warnings (not necessarily in consecutive games), you are suspended for the next game, and your warnings count goes back to zero. After the suspension, the rules are the same as at the beginning. You figure you will receive a warning after each game you play independently with probability p (0,).

4 8 POISSON PROCESS (a) Let the state of Markov chain be your warning count after a game. Write down the transition matrix and determine whether this chain is irreducible and aperiodic. Compute its invariant distribution. The transition matrix is P = p p 0 0 p p 0 0 and the chain is clearly irreducible (the transitions 0 0 happen with positive probability) and aperiodic (0 0 happens with positive probability). The invariant distribution is given by π 0 ( p) + π = π 0 π 0 p + π ( p) = π π p = π, and which gives π = π 0 + π + π =, [ ] p +, p +, p. p + (b) Write down an expression for the probability that you are suspended for both games 0 and 5. Do not evaluate. You have to have warnings after game 9 and then again warnings after game 4: P 9 0 P 4 0. (c) Let s n be the probability that you are suspended in the n th game. lim n s n. Compute As the chain is irreducible and aperiodic, lim s n = lim P(X n = ) = n n p + p.

5 8 POISSON PROCESS 3. A random walker on the nonnegative integers starts at 0, and then at each step adds either or 3 to her position, each with probability. (a) Compute the probability that the walker is at n + 3 after making n steps. The walker has to make (n 3) -steps and 3 3-steps, so the answer is ( ) n 3 n. (b) Let p n be the probability that the walker ever hits n. Compute lim n p n. The step distribution is aperiodic, as the greatest common divisor of and 3 is, so lim p n = n + 3 = A random walker is in one of six vertices, labeled 0,,,3,4, and 5, of the graph in the picture. At each time, she moves to a randomly chosen vertex connected to her current position by an edge. (All choices are equally likely and she never stays at the same position for two successive steps.) (a) Compute the proportion of times the walker spends at 0, after she makes many steps. Does this proportion depend on the walker s starting vertex? Independently of the starting vertex, the proportion is π 0, where [π 0,π,π,π 3,π 4 ] is the unique invariant distribution. (Unique because of irreducibility.) This chain is

6 8 POISSON PROCESS 3 reversible with invariant distribution given by 4 [4,,,,3,]. Therefore, the answer is 7. (b) Compute the proportion of times the walker is at an odd state (, 3, or 5) while she previously was at even state (0,, or 4). The answer is π 0 (p 03 + p 05 ) + π p + π 4 (p 43 + p 4 ) = = 5 4. (c) Now assume that the walker starts at 0. What is expected number of steps she will take before she is back at 0? The answer is π 0 = 7 5. In a branching process, an individual has two descendants with probability 3 4 and no descendants with probability 4. The process starts with a single individual at generation 0. (a) Compute the expected number of individuals in generation. As the answer is µ = = 3, µ = 9 4. (b) Compute the probability that the process ever goes extinct. As φ(s) = s

7 8 POISSON PROCESS 4 the solution to φ(s) = s is given by 3s 4s + = 0, i.e., 3s )(s ) = 0. The answer is π 0 = 3 6. Customers arrive at a two-server service stations, labeled and, as a Poisson process with rate λ. Assume time unit is one hour. Whenever a new customer arrives, any previous customers are immediately ejected from the system. A new arrival enters the service at station, then goes to station. (a) Assume the service time at each station is exactly hours. What proporition of entering customers will complete the service (before they are ejected)? The answer is P(customer served) = P(no arrival in 4 hours) = e 4λ. (b) Assume the service time at each station is now exponential with expectation hours. What proportion of entering customers will now complete the service? Now, P(customer served) = P( or more arrivals in rate Poisson process before one arrival in rate λ Poisson process) ) =( + λ = ( + λ). (c) Keep the service time assumption from (b). A customer arrives, but he is now given special treatment: he will not be ejected unless at least three or more new customers arrive during his service. Compute the probability that this special customer is allowed to complete his service. The special customer is served exactly when or more arrivals in rate Poisson

8 8 POISSON PROCESS 5 before three arrivals in rate λ Poisson process happen. Equivalently, among first 4 arrivals in rate λ+ Poisson process, or more belong to the rate one. The answer is ( ) 4 ) 3 λ λ λ + 4( λ + λ + = λ4 + λ 3 ( ) λ + 4.