Q3 Let N(t) be the number of failures of a computer system on the interval [0, t]. We suppose that {N(t),t > 0}is a Poisson process with rat A=1 per week.Q3 (i.) Calculate the probability that the system functions without failure during two consecutive weeks.Q3 (ii.) Calculate the probability that the system has exactly two failures during a given week, knowing it hasfunctioned without failure during the previous two weeks.Q3 (iii.) Calculate the probability that less than two weeks elapse before the third failure occurs.Q3(iv.) LetZ(t) = eN(t),t> 0.forShow thatE[Z(t]] = exp{t(e – 1)]using conditioning arguments.

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Answered: Q3 Let N(t) be the number of failures…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Could you solve (i), (ii), (iii)? Thank you.
3. Jobs arrive in a computer queue in the manner of a Poisson process with intensity λ. The central processor handles them one by one in the order of their arrival, and each has an exponentially distributed runtime with parameter , the runtimes of different jobs being independent of each other and of the arrival process. Let X (t) be the number of jobs in the system (either running or waiting) at time t, where X (0) = 0. Explain why X is a Markov chain, and write down its generator. Show that a stationary distribution exists if and only if λ < μ, and find it in this case.
Pedestrians approach a crossing from the left and right sides following independent Poisson processes withaverage arrival rates of λL = 5 and λR = 1 arrivals per minute. Each pedestrian then waits until a lightis flashed, at which time all waiting pedestrians must cross to the opposite side (either from left to right orfrom right to left). Assume that the left and right arrival processes are independent, that the light flashesevery T = 2 minutes, and that crossing takes zero time – it is instantaneous.1. What is the probability that in a particular crossing, there are total 10 pedestrian and they are allcrossing from left to right?
Could you solve (i), (ii), (iii)? Thank you.
Let i, denote the effective annual return achieved on an equity fund achieved between time (t-1) and time t. Annual log-returns on the fund, denoted by In(1+i), are assumed to form a series of independent and identically distributed Normal random variables with parameters u = 6% and σ = 14%. An investor has a liability of £10,000 payable at time 15. Calculate the amount of money that should be invested now so that the probability that the investor will be unable to meet the liability as it falls due is only 5%.
A Poisson process {N(t);t≥0} has a rate of λ=6 per hour. The events that occur are classified as type 1 and type 2. When an event occurs, there is a 0.3 probability that it is a type 1 event. Assuming the event types are independent of each other; find the probability that (i) not more than two events of type 1 and not fewer than four events of type 2 occur in any given three hour period.(round off answer to one decimal point)
Calls arrive at a switchboard according to a Poisson process with parameter λ = 5 per hour. (a) In a switchboard, what is the probability that it will be at least 15 minutes till the next call? (b) Find the probability that at least seven of the next ten waits between calls will be longer than 6 minutes. (c) Compute the probability that the first wait shorter than 6 minutes is the next wait.
R1
Two teams A and B play a soccer match. The number of goals scored by Team A is modeled by a Poisson process N₁ (t) with rate X₁ = 0.02 goals per minute, and the number of goals scored by Team B is modeled by a Poisson process N₂ (t) with rate A₂ = 0.03 goals per minute. The two processes are assumed to be independent. Let N(t) be the total number of goals in the game up to and including time t. The game lasts for 90 minutes. a. Find the probability that no goals are scored, i.e., the game ends with a 0-0 draw. b. Find the probability that at least two goals are scored in the game. c. Find the probability of the final score being Team A: 1, Team B: 2 d. Find the probability that they draw.
8 The failure of a circuit board interrupts work that utilizes a computing system, until a new board is delivered. The delivery time, Y, is uniformly distributed on the interval one to six days. The cost of a board failure and interruption includes the fixed cost co of a new board and a cost that increases proportionally to y². If C is the cost incurred, C = Co + c₂y². (a) Find the probability that the delivery time exceeds three days. (b) In terms of co and c₁, find the expected cost associated with a single failed circuit board.
Exercise 2. A r.v. X is Poisson distributed with parameter λ = 5.5. (a) Calculate the probability that X = 3. (b) Calculate the probability that X < 2. (c) What is the most likely (i.e. highest probability) value for X ? (You can make a graph of f(x) to find the answer).

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