随机过程2016期末考试及答案
合集下载
相关主题
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
随机过程2016期末考试及答案
Stochastic Processes Fall 2016 Final Exam: Gauss Award 26/12/16 Time Limit: 150 Minutes
Name (Print):
Advisor Name
This quiz contains 9 pages (including this cover page) and 8 problems. Check to see if any pages are missing. Enter all requested information on the top of this page, and put your initials on the top of every page, in case the pages become separated. You are required to show your work on each problem in this exam. The following rules apply:
Stochastic Processes
Final Exam: Gauss Award - Page 6 of 9
26/12/16
5. (20 points) A Yule process starts at time 0 with one organism. This organism splits into two organisms after a time Y1 with the density fY1 (y ) = λe−λy , y ≥ 0. Each of these two organisms splits into two more organisms after independent exponentially distributed delays, each with the same density λe−λy . In general, each old and new organism continues to split forever after a delay y with the same density λe−λy . (a) (5 points) Let T1 be the time at which the first organism splits, and for each i > 1, let Ti be the interval from (i − 1)st splitting until the ith. Show that Ti is exponential with parameter iλ and explain why the Ti are independent. (b) (5 points) For each n ≥ 1, let the continuous random variable Sn be the time at which the n th splitting occurs, i.e. Sn = T1 + . . . + Tn . Find a simple expression for the distribution function of Sn . (c) (5 points) Let X (t) be the number of organisms at time t > 0. Express the distribution function of X (t) for each t > 0 in terms of Sn for each n. Show that X (t) is a random variable for each t > 0 (i.e., show that X (t) is finite with probability 1). (d) (5 points) Find E [X (t)] for each t > 0.
Stochastic Processes
Final Exam: Gauss Award - Page 7 of 9
26/12/16
6. (20 points) Telephone calls arrive at a switching system in accordance with a Poisson process with rate of λ calls per minute. Every call independently lasts X minutes, where X is expo1 nentially distributed with E [X ] = µ minutes. Let N1 (t) and N2 (t) respectively, denote the numbers of completed and ongoing calls by the time t. Let N (t) = N1 (t) + N2 (t). (a) (5 points) What is the distribution of N2 (t)? (b) (5 points) Given that N (t) = n, what is the distribution of N2 (t)? (c) (10 points) Suppose that, upon the completion of each call, the system receives a revenue of X cents, where X is the call duration. Given that N (t) = n, what is the expected revenue of the system by the time t?
Final Exam: Gauss Award - Page 5 of 9
26/12/16
4. (20 points) A coin with probability p of Heads is flipped repeatedly. (a) (10 points) Suppose that p is a known constant, with 0 < p < 1. What is the expected number of flips until the pattern HTHT is observed? (b) (10 points) Now suppose that p is unknown, and that we use a Beta(a, b) prior to reflect our uncertainty about p (where a and b are known constants and are greater than 2). In terms of a and b, find the corresponding answers in this setting.
Stochastic Processes
Final Exam: Gauss Award - Page Leabharlann Baidu of 9
26/12/16
7. (20 points) Consider a population containing N copies of a gene that can each be one of two types, A or B . We model the number of genes of type A in successive generations, supposing that each generation has the same fixed number N of copies of the gene. Specifically, let Xn be the number of genes of type A in the nth generation, so Xn takes values in {0, 1, . . . , N }. Suppose a model of reproduction in which the population of genes at time n + 1 is obtained by drawing N times with replacement from the population at time n. (a) (5 points) Show that Xn is a Markov chain with transition probabilities given by p(i, j ) = (b) (5 points) Show that Yn = (c) (10 points) Show that N −1≤
Stochastic Processes
Final Exam: Gauss Award - Page 4 of 9
26/12/16
3. (20 points) Show the following inequalities. (a) (10 points) Let X be a Poisson random variable with rate λ. If there exists a constant a > λ, then P(X ≥ a) ≤ e−λ (eλ)a aa
(b) (10 points) Let X be a random variable with finite variance σ 2 . Then for any constant a > 0, 2σ 2 P(|X − E[X ]| ≥ a) ≤ 2 . σ + a2
Stochastic Processes
• Mysterious or unsupported answers will not receive full credit. A correct answer, unsupported by calculations, explanation, or algebraic work will receive no credit; an incorrect answer supported by substantially correct calculations and explanations might still receive partial credit. Do not write in the table to the right.
Problem 1 2 3 4 5 6 7 8 Total:
Points 15 15 20 20 20 20 20 20 150
Score
Stochastic Processes
Final Exam: Gauss Award - Page 2 of 9
26/12/16
1. (15 points) A coin that has probability of heads equal to p is tossed successively and independently until a head comes twice in a row or a tail comes twice in a row. Find the expected value of the number of tosses.
Stochastic Processes
Final Exam: Gauss Award - Page 3 of 9
26/12/16
2. (15 points) When three fair six-sided dice are rolled, what is the probability that the sum of the total numbers will be 12?
Stochastic Processes Fall 2016 Final Exam: Gauss Award 26/12/16 Time Limit: 150 Minutes
Name (Print):
Advisor Name
This quiz contains 9 pages (including this cover page) and 8 problems. Check to see if any pages are missing. Enter all requested information on the top of this page, and put your initials on the top of every page, in case the pages become separated. You are required to show your work on each problem in this exam. The following rules apply:
Stochastic Processes
Final Exam: Gauss Award - Page 6 of 9
26/12/16
5. (20 points) A Yule process starts at time 0 with one organism. This organism splits into two organisms after a time Y1 with the density fY1 (y ) = λe−λy , y ≥ 0. Each of these two organisms splits into two more organisms after independent exponentially distributed delays, each with the same density λe−λy . In general, each old and new organism continues to split forever after a delay y with the same density λe−λy . (a) (5 points) Let T1 be the time at which the first organism splits, and for each i > 1, let Ti be the interval from (i − 1)st splitting until the ith. Show that Ti is exponential with parameter iλ and explain why the Ti are independent. (b) (5 points) For each n ≥ 1, let the continuous random variable Sn be the time at which the n th splitting occurs, i.e. Sn = T1 + . . . + Tn . Find a simple expression for the distribution function of Sn . (c) (5 points) Let X (t) be the number of organisms at time t > 0. Express the distribution function of X (t) for each t > 0 in terms of Sn for each n. Show that X (t) is a random variable for each t > 0 (i.e., show that X (t) is finite with probability 1). (d) (5 points) Find E [X (t)] for each t > 0.
Stochastic Processes
Final Exam: Gauss Award - Page 7 of 9
26/12/16
6. (20 points) Telephone calls arrive at a switching system in accordance with a Poisson process with rate of λ calls per minute. Every call independently lasts X minutes, where X is expo1 nentially distributed with E [X ] = µ minutes. Let N1 (t) and N2 (t) respectively, denote the numbers of completed and ongoing calls by the time t. Let N (t) = N1 (t) + N2 (t). (a) (5 points) What is the distribution of N2 (t)? (b) (5 points) Given that N (t) = n, what is the distribution of N2 (t)? (c) (10 points) Suppose that, upon the completion of each call, the system receives a revenue of X cents, where X is the call duration. Given that N (t) = n, what is the expected revenue of the system by the time t?
Final Exam: Gauss Award - Page 5 of 9
26/12/16
4. (20 points) A coin with probability p of Heads is flipped repeatedly. (a) (10 points) Suppose that p is a known constant, with 0 < p < 1. What is the expected number of flips until the pattern HTHT is observed? (b) (10 points) Now suppose that p is unknown, and that we use a Beta(a, b) prior to reflect our uncertainty about p (where a and b are known constants and are greater than 2). In terms of a and b, find the corresponding answers in this setting.
Stochastic Processes
Final Exam: Gauss Award - Page Leabharlann Baidu of 9
26/12/16
7. (20 points) Consider a population containing N copies of a gene that can each be one of two types, A or B . We model the number of genes of type A in successive generations, supposing that each generation has the same fixed number N of copies of the gene. Specifically, let Xn be the number of genes of type A in the nth generation, so Xn takes values in {0, 1, . . . , N }. Suppose a model of reproduction in which the population of genes at time n + 1 is obtained by drawing N times with replacement from the population at time n. (a) (5 points) Show that Xn is a Markov chain with transition probabilities given by p(i, j ) = (b) (5 points) Show that Yn = (c) (10 points) Show that N −1≤
Stochastic Processes
Final Exam: Gauss Award - Page 4 of 9
26/12/16
3. (20 points) Show the following inequalities. (a) (10 points) Let X be a Poisson random variable with rate λ. If there exists a constant a > λ, then P(X ≥ a) ≤ e−λ (eλ)a aa
(b) (10 points) Let X be a random variable with finite variance σ 2 . Then for any constant a > 0, 2σ 2 P(|X − E[X ]| ≥ a) ≤ 2 . σ + a2
Stochastic Processes
• Mysterious or unsupported answers will not receive full credit. A correct answer, unsupported by calculations, explanation, or algebraic work will receive no credit; an incorrect answer supported by substantially correct calculations and explanations might still receive partial credit. Do not write in the table to the right.
Problem 1 2 3 4 5 6 7 8 Total:
Points 15 15 20 20 20 20 20 20 150
Score
Stochastic Processes
Final Exam: Gauss Award - Page 2 of 9
26/12/16
1. (15 points) A coin that has probability of heads equal to p is tossed successively and independently until a head comes twice in a row or a tail comes twice in a row. Find the expected value of the number of tosses.
Stochastic Processes
Final Exam: Gauss Award - Page 3 of 9
26/12/16
2. (15 points) When three fair six-sided dice are rolled, what is the probability that the sum of the total numbers will be 12?