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日期：2024-04-24 08:52

Stochastic Processes

Spring 2024 April 8, 2024

Homework #4: Exponential Distribution and Poisson Process

Due: April 25, noon

#1 (Chapter 5, Exercise 3) Let X be an exponential random variable. Without any

computations, tell which one of the following is correct. Explain your answer.

(a) E[X2|X > 1] = E[(X + 1)2]

(b) E[X2|X > 1] = E[X2]+1

(c) E[X2|X > 1] = (1 + E[X])2

#2 (Chapter 5, Exercise 4) Consider a post oce with two clerks. Three people, A,

B, and C, enter simultaneously. A and B go directly to the clerks, and C waits until either

A or B leaves before he begins service. What is the probability that A is still in the post

oce after the other two have left when

(a) the service time for each clerk is exactly (nonrandom) ten minutes?

(c) the service times are exponential with mean 1/μ?

#3 (Chapter 5, Exercise 50) The number of hours between successive train arrivals

at the station is uniformly distributed on (0, 1). Passengers arrive according to a Poisson

process with rate 7 per hour. Suppose a train has just left the station. Let X denote the

number of people who get on the next train. Find

(a) E[X],

(b) V ar(X).

#4 (Chapter 5, Exercise 59) Cars pass an intersection according to a Poisson process

with rate . There are 4 types of cars, and each passing car is, independently, type i with

probability pi,

P4

i=1 pi = 1.

(a) Find the probability that at least one of each of car types 1, 2, 3 but none of type 4 have

passed by time t.

(b) Given that exactly 6 cars of type 1 or 2 passed by time t , find the probability that 4 of

them were type 1.

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#5 Numerical Experiment (Simulation of Poisson Process) In the lecture, we

have introduced two ways to simulate a Poisson processes. In this exercise, you need to

implement the two approaches via Python and validate that your code is correct.

In the attachment, you will find codes to generate exponential, Poisson and uniform

distributions. You can use those codes to implement your simulation. Besides, it also

contains codes to plot exponential and Poisson distribution functions. You can use those

codes to validate your simulation results.

1. Write two functions Poisson1 and Poisson2 to simulate the sequence of arrival times

of events on [0, 1] for a Poisson process with rate , where  is the input to your

functions. In Poisson1, the simulation is based on generating i.i.d. inter-arrival times.

In Poisson2, you first generate the total number of arrivals and then the conditional

distribution of arrival times.

2. Set  = 10. Run Poisson1 for 10000 rounds, record the total number of arrivals in each

round. Plot the empirical distribution of the simulated number of arrivals, and validate

your codes by comparing the empirical distribution with the theoretic distribution.

3. Set  = 10. Run Poisson2 for 10000 rounds, record the first arrival time. (What if

there is no arrival on [0, 1]?) Plot the empirical distribution of the first arrival time,

and validate your codes by comparing the empirical distribution with the theoretic

distribution.

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