Arrivals of the Number 1 bus at a particular bus stop form a Poission process with...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Arrivals of the Number 1 bus at a particular bus stop form a Poission process with rate parameter X, and arrivals of the Number 2 bus at the same bus stop form a Poisson process with rate parameter X₂. The two Poisson processes are independent. (a) What is the probability that precisely 4 buses (of type Number 1 and/or Number 2) arrive in the time interval [0, t]? (b) What is the probability that exactly 3 Number 2 buses pass while I am waiting for a Number 1 bus? (c) When the maintenance depot goes on strike half the buses break down before they reach my stop. What then is the probability that not a single bus passes in the time interval [0.t]. When answering each of parts (a), (b) and (c) you should briefly explain your reasoning. Hints: (i) Recall that 5² 2² -le-Br dr е I'(a) 30 where the gamma function I(-) is such that, for a positive integer n, I(n) = (n − 1)! (ii) In part (c) you may assume that thinning has taken place, as defined in the lecture notes. Question 2. [10 marks]. Daleks from the planet Skaro are invading Earth to master the planet for their own use. The creator of the Dalek race, Davros, sent out N Daleks to the battlefield on earth to attack humans. Here N is a Poisson random variable with mean X. The earth has its own defence system put up by the Torchwood Institute, which specialises in alien weaponry. Some Daleks in the battle field can be shot down by humans. Assume each Dalek dies independently in the battlefield. Let X; be the lifetime of the ith Dalek in the battlefield, which is a positive random variable with PDF f and CDF F. Let N(t) be the number of Daleks who have died by time t. (a) Find the mean of N (t) for each t > 0. (b) Identify the distribution of N(t) for each t > 0. (c) Show that for any 0 ≤ t₁ < t₂ ≤ t3 < ₁, N(t₂) - N(t₁) and N(₁) - N(t3) are independent. What type of process is N(t)? (d) Let Y; be a Gamma random variable with parameter (a, 3). Assume (Y;) are indepen- dent. Define a compound Poisson process N(t) Y(t) = ΣΥ. i=1 Find E[Y (t)] where t > 0. (e) In the context of part (d), find Cov (Y(s), Y(s+t)) for each s, t > 0. = Question 3. [12 Marks]. Consider a Markov process {X(t)}20 with state space S {0, 1,2,3} and Q-matrix, or genarator, -90 2 0 0 2 -91 0 0 0 0 2 -92 1 -93, Determine the following quantities. (a) The expected holding time of each state (i.e. the expected amount of time spent in each state before a jump). (b) The stationary distribution TQ associated with Q. (c) The proportion of time the process spends in state 3 in the long run. (d) The expected return time, m₂, for state 2. (e) The transition matrix R of the embedded chain. (f) A stationary distribution of the embedded chain. Is the stationary distribution TR that you found unique? Explain your reasoning. de-At Question 4. [9 Marks]. Recall that the exponential PDF has the form f(t; X) where t, > > 0 and X is known as the rate parameter. Consider a job shop that consists of 3 identical machines and 2 technicians. Suppose that, the amount of time each machine operates before breaking down is exponentially distributed with rate parameter 0.1 and, a technician takes to fix a machine is exponentially distributed with rate parameter 0.4. Sup- pose that all the times to breakdown and times to repair are independent random variables and let X(t) be the number of machines which are operating at time t. (a) Determine the Q-matrix for this Markov process. (b) Write the forward equations involving the Poj(t), j = 0, 1,2, 3, in terms of Pj(t)= Poj(t) = P(X(t) = j|X (0) = 0). (c) Obtain the equilibrium probabilities p;= lim Pj (t). t-∞ = (d) What is the average number of busy technicians in the long-run? [Note: do not expect your answer to be an integer.] Arrivals of the Number 1 bus at a particular bus stop form a Poission process with rate parameter X, and arrivals of the Number 2 bus at the same bus stop form a Poisson process with rate parameter X₂. The two Poisson processes are independent. (a) What is the probability that precisely 4 buses (of type Number 1 and/or Number 2) arrive in the time interval [0, t]? (b) What is the probability that exactly 3 Number 2 buses pass while I am waiting for a Number 1 bus? (c) When the maintenance depot goes on strike half the buses break down before they reach my stop. What then is the probability that not a single bus passes in the time interval [0.t]. When answering each of parts (a), (b) and (c) you should briefly explain your reasoning. Hints: (i) Recall that 5² 2² -le-Br dr е I'(a) 30 where the gamma function I(-) is such that, for a positive integer n, I(n) = (n − 1)! (ii) In part (c) you may assume that thinning has taken place, as defined in the lecture notes. Question 2. [10 marks]. Daleks from the planet Skaro are invading Earth to master the planet for their own use. The creator of the Dalek race, Davros, sent out N Daleks to the battlefield on earth to attack humans. Here N is a Poisson random variable with mean X. The earth has its own defence system put up by the Torchwood Institute, which specialises in alien weaponry. Some Daleks in the battle field can be shot down by humans. Assume each Dalek dies independently in the battlefield. Let X; be the lifetime of the ith Dalek in the battlefield, which is a positive random variable with PDF f and CDF F. Let N(t) be the number of Daleks who have died by time t. (a) Find the mean of N (t) for each t > 0. (b) Identify the distribution of N(t) for each t > 0. (c) Show that for any 0 ≤ t₁ < t₂ ≤ t3 < ₁, N(t₂) - N(t₁) and N(₁) - N(t3) are independent. What type of process is N(t)? (d) Let Y; be a Gamma random variable with parameter (a, 3). Assume (Y;) are indepen- dent. Define a compound Poisson process N(t) Y(t) = ΣΥ. i=1 Find E[Y (t)] where t > 0. (e) In the context of part (d), find Cov (Y(s), Y(s+t)) for each s, t > 0. = Question 3. [12 Marks]. Consider a Markov process {X(t)}20 with state space S {0, 1,2,3} and Q-matrix, or genarator, -90 2 0 0 2 -91 0 0 0 0 2 -92 1 -93, Determine the following quantities. (a) The expected holding time of each state (i.e. the expected amount of time spent in each state before a jump). (b) The stationary distribution TQ associated with Q. (c) The proportion of time the process spends in state 3 in the long run. (d) The expected return time, m₂, for state 2. (e) The transition matrix R of the embedded chain. (f) A stationary distribution of the embedded chain. Is the stationary distribution TR that you found unique? Explain your reasoning. de-At Question 4. [9 Marks]. Recall that the exponential PDF has the form f(t; X) where t, > > 0 and X is known as the rate parameter. Consider a job shop that consists of 3 identical machines and 2 technicians. Suppose that, the amount of time each machine operates before breaking down is exponentially distributed with rate parameter 0.1 and, a technician takes to fix a machine is exponentially distributed with rate parameter 0.4. Sup- pose that all the times to breakdown and times to repair are independent random variables and let X(t) be the number of machines which are operating at time t. (a) Determine the Q-matrix for this Markov process. (b) Write the forward equations involving the Poj(t), j = 0, 1,2, 3, in terms of Pj(t)= Poj(t) = P(X(t) = j|X (0) = 0). (c) Obtain the equilibrium probabilities p;= lim Pj (t). t-∞ = (d) What is the average number of busy technicians in the long-run? [Note: do not expect your answer to be an integer.]
Expert Answer:
Answer rating: 100% (QA)
when two processes are independent the combined rate of parameters is the sum of two individual para... View the full answer
Related Book For
Statistics For Engineering And The Sciences
ISBN: 9781498728850
6th Edition
Authors: William M. Mendenhall, Terry L. Sincich
Posted Date:
Students also viewed these mathematics questions
-
Managing Scope Changes Case Study Scope changes on a project can occur regardless of how well the project is planned or executed. Scope changes can be the result of something that was omitted during...
-
Planning is one of the most important management functions in any business. A front office managers first step in planning should involve determine the departments goals. Planning also includes...
-
You have an opportunity to choose a flight for your upcoming spring break trip to Mexico. After a lot of thought and research, you have narrowed your options to four different flights. If there are...
-
International Paper reported the following items for the current year: Sales = $3,500,000; Cost of Goods Sold = $1,250,000; Depreciation Expense = $170,000; Administrative Expenses = $150,000;...
-
1. Taxable tips 2. Personal allowance 3. Backup withholding 4. Form 1099-MISC 5. Roth IRA 6. Standard deduction 7. Pretax salary reductions 8. Form 1096 9. Nontaxable fringe benefits 10. Flexible...
-
Formulate a determinantal criterion similar to Exercise 3.5.15 for negative definite marices. Write out the 2 x 2 and 3 x 3 cases explicitly.
-
Cotton, Corp., uses the allowance method to account for uncollectible accounts. On May 31,2010, Allowance for Uncollectible Accounts has a $1,300 credit bal ance. Journalize the year-end adjusting...
-
BPO Services is in the business of digitizing information from forms that are filled out by hand. In 2006, a big client gave BPO a distribution of the forms that it digitized in house last year, and...
-
Investigate the interdisciplinary approaches of advanced resilient urban planning and design, incorporating principles of climate resilience, green infrastructure, and community engagement to address...
-
For the case The WM. Wrigley Jr. Company: Capital Structure, Valuation, and the Cost of Capital(Darden Case: UVAF1482) please answer the following questions and explain your reasoningwhere...
-
Mary is ordering flowers to send to her mom for Mother's Day. Each flower she orders costs $2 each and she also has to pay a $4 delivery fee. If Mary's total bill is $18, how many flowers will Mary's...
-
Wildhorse Company is considering two different, mutually exclusive capital expenditure proposals. Project A will cost $ 4 6 6 , 0 0 0 , has an expected useful life of 1 3 years and a salvage value of...
-
As a tax professional, understanding the key components associated with taxation is crucial for ensuring accurate financial reporting and compliance. Accounting profit represents the profit reported...
-
Your team is creating a small consulting firm to help small regional banks implement risk-adjusted return on capital (RAROC) initiatives. Your first assignment is coming from a motor financing...
-
George purchased a life annuity for 6 , 2 0 0 that will provide him $ 1 5 5 monthly payments for as long as he lives. Based on IRS tables, georges life expectancy is 1 0 0 months. How much of the...
-
Edwin Hubble's historical estimate for the Hubble constant, H0, was significantly different from our currently favored values, which are near 70 km/s/Mpc. Based on analysis of his own data, Hubble...
-
The inability of a pilot to carry out the duties required to fly an aircraft is sometimes called _________. orientation incapacitation discombobulation alienation
-
What tools are available to help shoppers compare prices, features, and values and check other shoppers opinions?
-
In Reliability Engineering and System Safety (Jan. 2006), nuclear and quantum engineers at the Korea Advanced Institute of Science and Technology designed a digital safety system for testing system...
-
Refer to Exercise 6.12. a. Find E (Y - 10). b. Find E(3Y). Data from Exercise 6.12 The Department of Transportation (DOT) monitors sealed bids for new road construction. For new access roads in a...
-
A company is studying three different safety programs, A, B, and C, in an attempt to reduce the number of work-hours lost because of accidents. Each program is to be tried at three of the companys...
-
For each of the following events concerning disclosure of events that took place after year end, discuss the manner in which it should be disclosed in the financial statements or the audit report....
-
Determining whether a subsequent event requires an adjustment to the financial statements is difficult in many circumstances. Discuss the primary difference between subsequent events that require...
-
Each of the following techniques for managing earnings was described in the chapter: - "Big Bath" charges - Write-off of acquired assets - "Cookie Jar" reserves - Abuse of materiality - Questionable...
Study smarter with the SolutionInn App