Assume that N(t) is a Poisson process with rate λ=2024. ComputeP(N(s) =1000,N(t) =2023),for any s≤t
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Assume that N(t) is a Poisson process with rate λ=2024. ComputeP(N(s) =1000,N(t) =2023),for any s≤t
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.Suppose the customers arrive at a Poisson rate of on eper every 12 minutes, and that the service time is exponential at a rate of one service per 8 minutes. What are the average number of customers in the system(L) and the average time a customer spends in the system(W)?Suppose the customers arrive at a Poisson rate of on eper every 12 minutes, and that the service time is exponential at a rate of one service per 8 minutes. What are the average number of customers in the system(L) and the average time a customer spends in the system(W)? Now suppose that the arrival rate increases 20 percent.What is the corresponding change in L and W?
- 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?The number of accidents incurred by insured drivers in a given day (24 hours, starting from 6:00am) is a Poisson process with rate function 1+½t, 4, if 0 < t < 4, if 4 < t < 12, if 12 ≤ t ≤ 16, X(t) = 12t, 2, if 16 ≤ t ≤ 24 If an accident occurs, the probability that the damage amount exceeds the de- ductible is 0.25. Suppose that the amount of each claim is uniformly distributed on [$2000, $5000] and all claims are independent. (a) Find the probability that there are no accidents that are required to claim during 5pm to 8pm. (b) Find the average total amount of the claims requested in 5 hours from 8am. (c) Let N be the number of claims with claim amount less than $3000 during 10am to 2pm. Find the distribution of N. (d) It is observed that the accidents comes from seniors with probability 2/3 during 11am to 4:00pm and with probability 1/3 during other time period. Let Xt denote the number of accidents incurred by insured seniors drivers at t hours starting from 6:00am. Explain why…Let {Nt}{t > 0} be a Poisson process with rate l=2. Find the probability that we see 3 events in the interval [0,3) and 2 events in the interval (1,6]. Round answer to 5 decimals.
- b and c part Buses arrive at the station according to a Poisson process at a rate of lambda = 0.4 per minute. Assume at the starting condition is time = 0. Imagine when a bus stops, the probability 1 person gets off at the bus station is p = 0.7, and the probability 2 people get off is p = 0.3, independent of everything else. Let X denote the number of people that get off at the bus station in the first 5 minutes. a) Find E[X] b) Calculate P{X = 2} c) Compute Var(X)Suppose that the random change in value of a financial asset is X over the first day and Y over the second. Suppose also that Var(X) =18 and Var(Y) = 26 In this case, the total change in the value over these two days is given by X +Y. Do you have enough information to compute Var(X +Y)? If so, compute this value. If not, explain what additional information you need to do so.In an insurance company the number of claims are modelled as a Poisson process with rate λ > 0. Assume that the size of all claims is a fixed amount a > 0, the initial surplus is denoted by u, with 0 < u < a. If the premium income per unit time is 1.73λa, find the probability that ruin happens at the first claim.
- Let (x(t)) be a Poisson process with rate A, and let T₁ = min{t: X(t) > 1} be the first arrival time. Prove that T₁ is exponentially distributed with rate \.= Customers arrive at a certain facility according to a Poisson process of rate > 5 customers/hour. Suppose that it is known that 10 customers arrived in the first 3 hours. Determine the mean total waiting time E[W₁ + W₂+ · + W₁0 | N(3) = 10] : = 55/15 hoursCustomers arrive at a service facility according to a Poisson process of rate λ = 7 customers/hour. Let N(t) be the number of customers that have arrived up to time t hours. Let W₁, W2, W3, ... be the successive arrival times of the customers. (a) Find the expected arrival time of the 10-th customer, E[W₁0] : = hours (b) Given N(3) = 3, determine the expected arrival time of the 10-th customer, E[W₁0 | N(3) = 3] = hours.