Question 2(2.1) Prove the following: If Poisson process X, with rate A counts the arrivals of type 1 customersμinto a shop, and if Poisson process Y, with rate counts the arrivals of type 2 customers intothe shop, and if the arrival times of the two types of customers are independent of each other,then the process Z, counting the total number of arrivals is also a Poisson process. What isthe rate of the process Z?(2.2) Assume that a customer relations officer receives nice e-mails at the rate of 2 per hour, andnasty e-mails at the rate of 1 per hour. The e-mails all arrive independent of each other.Calculate the probabilities of the following events:(a) He receives no e-mails at all during a working day of 8 hours.(b) He receives only nice e-mails during the next 4 hours.(c) The next 3 e-mails are all nasty.

Step 1:2.1(a).Given data :Xt is a Poisson process with rate λ, counting arrivals of type 1…

Answered: Question 2 (2.1) Prove the following:…

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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9. Consider a bank office where customers arrive according to a Poisson process with an average arrival rate of customers per minute. The bank has only one teller servicing the arriving customers. The service time is exponentially distributed and the mean service rate is μ customers per minute. It turns out that the customers are impatient and are only willing to wait in line for an exponential distributed time with a mean of 1/μ minutes. Assume that there is no limitation on the number of customers that can be in the bank at the same time. a. Construct a rate diagram for the process and determine what type of queuing system this correspond to on the form A1/A2/A3. b. Determine the expected number of customers in the system when λ = 1 and μ = 2. c. Determine the average number of customers per time unit that leave the bank without being served by the teller when λ = 1 and µ = 2.
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?
Figure 1 shows a sample path of a Poisson process where the sequence of random variables, {S1, S2, ...}, show the arrival times and the sequence, {X1, X2, ...}, consists of the interarrival times that are i.i.d. exponential random variables with E[X;] = 1/A. %3D 3. Š, Š, S, S, Figure 1: A sample path of a Poisson process. nd A such that S = AX where X = [X1 X2 X3 X4]T and S = [S1 S2 S3 Sa]™. Find fs(s) using fx(x) and A (a) (b) Hint: You may take the determinant of A as 1. 2.
2. In a different warehouse, book orders are handled by two members of staff, A and B, who take turns to pack each orders as it arrives (this is a non-random alternation). Assume orders arrive as a Poisson process with rate A per hour, that A packs the first order, and that orders are packed instantaneously. Let NA(t) be the number of orders packed by A by time t, and let X₁, X2.... be the inter-arrival times for the process (NA()). (a) What is the probability distribution for X.? Hint: relate the X, to the times between orders. (b) Is (NA(t)) a Poisson process? Explain your reasoning.
b) Let X₁, X₂,..., X and Y₁, Y₂, ..., Ym be random samples from populations with moment generating 25 functions Mx₁(t) = ³t+t² and My(t) = (₁-¹)²5, respectively. i) Find the sampling distribution of the statistic W = X₁ + 2X₂ − X3 + X4 + X5. ii) What is the value of the sample size n, if P[Σ1(X¡ − X)² > 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Ỹ - µy| ≥10) < 0.04?
1. (a) lind autocovariance function and autocorrelation function of Poisson Process. (b) If X(1) = A cos ct+ Bsin et and Y) and B are uncorrelated random variables with mean 0 and variance 1. Examine whether {X(1)} and {Y(!)} are jointly widen -sense stationary or not. Bcoset - Asin ct. where c is a constant, A
Question 5. A professor decides to run an experiment to measure the effect of time pressure on final exam scores. He gives each of the 400 students in his course the same final exam, but some students have 90 minutes to complete the exam while others have 120 minutes. Each students is randomly assigned one of the examination times based on the flip of a coin. Let Y, denote the number of points scored on the exam by the ith student (0 ≤ Y ≤ 100), let X, denote the amount of time that the student has to complete the exam (X, = 90 or 120), and consider the regression model Yi = Bo + BiXitu. (1) Explain what the term u, represents. Why will different students have different values of ui? (2) Explain why E(u₁|X₁) = 0 for this regression model. (3) The Least Squares Assumptions Yi Bo + B₁Xi + Ui, i = 1,..., n, where I The error term u, has conditional mean zero given X₂: E(ui Xi) = 0; II (X₁, Yi), i 1,, n, are independent and identically distributed (i.i.d.) draws from their joint…
In time-series decomposition, seasonal factors are calculated by Multiple Choice O O O O O SFt (Y) (CMA). SFt= Y/CMAt (CMA+) x (SFt) =Yt. SFt = Yt - CMAt. None of the options are correct.
Which of the following statements are true (no need to justify)?(1) The transformation of a continuous random variable is always a continuous random variable.(2) The distribution of the transformation of a random variable is always the same as that of the original sat-distribution of the univariate.(3) Many random variables can be represented by a uniformly distributed random variableusing if the quantile function of the distribution is known.(4) The accumulation function of several variables is increasing with respect to each variablefunction.(5) The multidimensional accumulation function is determined by its (one-dimensional) marginal distributionwhere from.(6) The marginal distributions of a discrete random vector are discrete distributions.
Define the random process {Xt} by Xt = et + θ et−1. Show this process is weakly stationary.
3
b) Let X₁, X₂, X, and Y₁, Y₂,...,Y be random samples from populations with moment generating 25 functions Mx, (t) = e³t+t² and My(t) = respectively. i) Find the sampling distribution of the statistic W = X₁ + 2X₂-X₂ + X₁ + X5. ii) What is the value of the sample size n, if PIX(X-X)² > 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|-|≥10) < 0.04?

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