2. In a different warehouse, book orders are handled by two members of staff, A and B, who taketurns to pack each orders as it arrives (this is a non-random alternation). Assume orders arriveas a Poisson process with rate A per hour, that A packs the first order, and that orders arepacked instantaneously.Let NA(t) be the number of orders packed by A by time t, and let X₁, X2.... be the inter-arrivaltimes 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.

Given that, In a warehouse, book orders are handled by two members of staff, staff A and staff B.…

Answered: 2. In a different warehouse, book…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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We consider a group of 4 students among which a rumour is spreading. At time 0, only one studentis aware of the rumour. Whenever two students meet, if one is aware of the rumour and not theother, the second one becomes aware of it. We assume that for any pair of students, the times atwhich they meet forms a Poisson process with rate 1. We also assume that, for the 6 possible pairsof students, we get 6 independent Poisson processes.a. Let A(t) be the number of students aware of the rumour at time t. We admit this is a continuous-time Markov chain. Give its parameters. No proof is required.b. Let T be the first time at which all students become aware of the rumour. Compute E[T ]
1) Classify by defining the ndex sets and state spaces of random processes given below. (a). For t>0, the water level msrde a tanter up to t tme, (b). The water level mside a tanker at the end of each hour. (6). For t20, the number of customers wartng in line at a bank t time, (d). The number of farlures of a machme per week,
Consider a (non)-symmetric random walk M = Xo +X1+X2+…+Xn where X;'s (i> 1) are independent and identical distributed, P(X; = 1) = p and P(X;=-1) =1–p and Xo = 0. Let Sn = eaS,-n Denote Fn= 0(X1,...,Xn). Find a such that (Sn)n20 is a (Fn)n20 - martingale.
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Gg.212.
If X is continaonsly and uniformly distributerd between and 10. What is the Select one: a.0.6 E None of these e.08
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.
Let {e} be a zero-mean, unit-variance white noise process. Consider a process {Y} which starts at time t = 0, and is defined recursively as follows: Let Yo = cieo, Y₁ = c₂Yo + e₁, and Y₁ = ¢₁Yt-1 + 2Yt-2 + et for t≥ 2 (as in an AR(2) process). a) Show that the process has zero mean. 1 b) For values of 01, 02 in the stationarity region for an AR (2) process, show how to choose C₁, C₂ such that both Var(Y) = Var(Y₁) and the lag 1 autocorrelation cov(Yo, Y₁) equals that of a stationary AR(2) process with parameters 01, 02. c) Once the process {Y} has been generated, show how one can transform in into a new process which has any desired mean and variance. (Remark. This gives a method for simu- lating stationary AR(2) processes.)
4
Define the random process {Xt} by Xt = et + θ et−1. Show this process is weakly stationary.
Prove the following
Digi-Key is a large distributor (but not manufacturer) of electronic components. They receive shipments of components from manufacturers and sell them on to other manufacturers, consumers, and hobbyists. An electrical fuse is generally a small device meant to protect other electrical components from damaging surges of electrical current. Suppose Digi-Key receives shipments of a particular fuse in boxes of 20,000 fuses. The manufacturer of this fuse claims that just 1 out of every 100 fuses is defective. Digi-Key draws 50 fuses randomly from each box and tests them using an error-free method. If it finds 2 or more defective fuses in its sample of 50 fuses, it sends that box back to the manufacturer and gets a replacement box. (Because there are 20,000 fuses in a box--much bigger than 50--you may treat this as if it is sampling with replacement even though it is not really.) Compute the probability that Digi-Key sends a box back to the manufacturer if the manufacturer's claim (that…

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