I cycle for 5 miles along the road. What is the expectation of the number ofpotholes I pass?A repair crew travel along the road, repairing each pothole they pass until theyhave repaired 10 potholes. What is the expectation of the distance they travel? The potholes in a long road occur as a Poisson process of rate 4 per mile. In otherwords, if we letR(t) = the number of potholes in the first t miles of the roadthen the random variables (R(t): t > 0) form a Poisson process of rate 4.

Given that, The number of miles I cycled is miles.The rate of potholes per mile is .The number of…

Answered: The potholes in a long road occur as a…

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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At 15:00 it is the end of the school day, and it is assumed that the departure of the students from school can be modelled by a Poisson distribution. On average, 24 students leave the school every minute. (e) There are 200 days in a school year. Given that Y denotes the number of days in the year that at least 700 students leave before 15:30, find (ii) P(Y > 150).
Consider a linear birth–death process where the individual birth rate is λ=1, the individual death rate is μ= 3, and there is constant immigration into the population according to a Poisson process with rate α. Please explain and show work! (a) State the rate diagram and the generator. (b) Suppose that there are 10 individuals in the population. What is the probability that the population size increases to 11 before it decreases to 9? (c) Suppose that α = 1 and that the population just became extinct. What is the expected time until it becomes extinct again? α in Greek letter alpha.
Suppose that earthquakes occur according to a Poisson process with rate λ = 0.2 per day. (a) What is the probability that there will be 2 or fewer earthquakes in April? (b) Starting from now, what is the probability that the second earthquake will occur within a week?
The random variable X follows a Poisson process with the given value of and t. Assuming A= 0.12 and t= 11, compute the following. (a) P(4) (b) P(X < 4) (c) P(X 2 4) (d) P(3 sXs7) (e) Hx and oy (a) P(4) 0.0338 (Do not round until the final answer. Then round to four decimal places as needed.) (b) P(X < 4) = 0.9549 (Do not round until the final answer. Then round to four decimal places as needed.) (c) P(X2 4) 0.0451 (Do not round until the final answer. Then round to four decimal places as needed.) (d) P(3 3XS7) (Do not round until the final answer. Then round to four decimal places as needed.) (e) Hx (Round to two decimal places as needed.) oy (Rasund to three decimal places as needed.)
Let Y be a Poisson random variable with mean λ = 2. (a) Find P(Y ≥ 2). (b) Find P(Y ≥ 4|Y ≥ 2).
The US divorce rate has been reported as 3.6 divorces per 1000 population. Assuming that this rate applies to a small community of just 500 people and it’s Poisson distributed, and that x = the number of divorces in this community during the coming year, determine the following : a. E(x) b. P(x=1) c. P(x=4) d. P(x6) e. P(2x5)
A facility produces items according to a Poisson process with rate λ. However, it has shelf space for only k items and so it shuts down production whenever k items are present. Customers arrive at the facility according to a Poisson process with rate μ. Each customer wants one item and will immediately depart either with the item or empty handed if there is no item available. (a) Find the proportion of customers that go away empty handed. (b) Find the average time that an item is on the shelf. (c) Find the average number of items on the shelf.
If it is known that the random process X (t) is a stationary process in a broad sense, could the following autocorrelation function belong to this process? Explain the reason for your answer.
Moore’s portable X-ray equipment repair service shop receives an average of 6 portable X-ray sets per 8-hour day to be repaired. Moore would like to be able to tell his customers that they can expect one day service. What average repair time per set will the repair shop have to achieve to provide 1-day service on average? Assume that the arrival rate is Poisson distributed and repair times are exponentially distributed.
Given two independent Poisson processes of rates λ1 and λ2, what is the probability that process 1 has more arrivals than process 2 in 1 second?
The arrival of customers to a small local food restaurant may be modeled by a Poisson process with rate of 1 per 15 min period. Assume customer will only arrive if there is at least one customer present. Each customer on average stays for a time exponentially distributed with mean 20 minutes. This is modeled as a birth-death process. (i) Compute the probability that the capacity of 20 will be reached if we start with 5 customers.
If the number X of particles emitted during a 1-h period from a radioactive source has a Poisson distribution with parameter 2 = 4 and that the probability that any emitted particle is recorded is p = 0.9, find the probability distribution of the number Y of the particles recorded in a 1-h period and hence the probability that no particle is recorded.

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The potholes in a long road occur as a Poisson process of rate 4 per mile. In other words, if we let R(t) = the number of potholes in the first t miles of the road then the random variables (R(t): t > 0) form a Poisson process of rate 4.