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Prob pp2

The document contains 9 questions about Poisson processes. The questions cover topics like the characteristics of Poisson processes, the probability of certain numbers of customers arriving over time periods based on arrival rates, and the mean and variance of purchases made over time based on a Poisson arrival process and geometric purchase amounts.

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Mohi Gpt4
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32 views2 pages

Prob pp2

The document contains 9 questions about Poisson processes. The questions cover topics like the characteristics of Poisson processes, the probability of certain numbers of customers arriving over time periods based on arrival rates, and the mean and variance of purchases made over time based on a Poisson arrival process and geometric purchase amounts.

Uploaded by

Mohi Gpt4
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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PROBLEM SHEET 4: POISSON PROCESSES

Question 6
Suppose that cars arrive at the petrol station according to a Poisson process,
{Nt }t≥0 of rate λ ∈ R+ . In addition, independently, a car is green with probability
p; let {Ntg }t≥0 denote the number of green cars that have appeared. Show that
{Ntg }t≥0 is a Poisson process of rate λp.

Question 7
Let {Nt } be a Poisson process of rate λ ∈ R+ and X1 , X2 , . . . be a sequence of
i.i.d random variables, such that their characteristic function exists. Further {Nt }
and {Xi } are independent. Let
Nt
X
Zt = Xi .
i=1
Find the Characteristic function of Zt , t > 0.

Question 8
Customers arrive at a bank according to a Poisson process at a mean rate of
λ = 10 per minute. 60% of the customers wish to withdraw money (type A), 30%
wish to pay in money (type B), and 10% wish to do something else.
(a) What is the probability that more than 5 customers arrive in 30 seconds?
(b) What is the probability that in 1 minute, 6 type A customers, 3 type B cus-
tomers, and 1 type C customers arrive?
(c) If 20 customers arrive in 2 minutes, what is the probability that just one wants
to carry out a type C transaction?
(d) What is the probability that the first 3 customers arriving require only to make
a type A transaction?
(e) How long a time will elapse until there is a probability of 0.9 that at least one
customer of type A and one of type B will have arrived? (you will need to solve
this numerically).

Question 9
A bank opens at 10.00am and customers arrive according to a non-homogeneous
Poisson process at a rate 10(1 + 2t), measured in hours, starting from 10.00.
(a) What is the probability that two customers have arrived by 10.05?
(b) What is the probability that 6 customers arrive between 10.45 and 11.00?
(c) What is the probability that more than 50 customers arrive between 11.00 and
12.00?
(d) What is the median time to the first arrival after the bank opens?
(e) By what time is there a probability of 0.95 that the first customer after 11.00
will have arrived?
1
2 PROBLEM SHEET 4

Question 10
A person makes shopping expeditions according to a Poisson process with rate
λ ∈ R+ . The number of purchases he makes is distributed according to a geometric
distribution Geo(p). What are the mean and variance of the number of purchases
made in time t?

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