Page 40 - A First Course In Stochastic Models

EXERCISES 31

that the total amount claimed during a given year has a negative binomial distribution with
parameters −λ/ ln(1 − α) and 1 − α.
1.20 An insurance company has two policies with fixed remittances. Claims from the policies
1 and 2 arrive according to independent Poisson processes with respective rates λ 1 and λ 2 .
Each claim from policy i is for a fixed amount of c i , where c 1 and c 2 are positive integers.
Explain how to compute the probability distribution of the total amount claimed during a
given time period.
1.21 It is only possible to place orders for a certain product during a random time T which
has an exponential distribution with mean 1/µ. Customers who wish to place an order
for the product arrive according to a Poisson process with rate λ. The amounts ordered
by the customers are independent random variables D 1 , D 2 , . . . having a common discrete
distribution {a j , j = 1, 2, . . . }.
2
(a) Verify that the mean m and the variance σ of the total amount ordered during the
random time T are given by
λ 2 λ 2 λ 2 2
m = E(D 1 ) and σ = E(D ) + E (D 1 ).
1
µ µ µ 2
(b) Let {p k } be the probability distribution of the total amount ordered during the random
time T . Argue that the p k can be recursively computed from
k
λ
p k = p k−j a j , k = 1, 2, . . . ,
λ + µ
j=1
starting with p 0 = µ/(λ + µ).
1.22 Consider a non-stationary Poisson arrival process with arrival rate function λ(t). It is
assumed that λ(t) is continuous and bounded in t. Let λ > 0 be any upper bound on the
function λ(t). Prove that the arrival epochs of the non-stationary Poisson arrival process can
be generated by the following procedure:
(a) Generate arrival epochs of a Poisson process with rate λ.
(b) Thin out the arrival epochs by accepting an arrival occurring at epoch s with probability
λ(s)/λ and rejecting it otherwise.
1.23 Customers arrive at an automatic teller machine in accordance with a non-stationary
Poisson process. From 8 am until 10 am customers arrive at a rate of 5 an hour. Between
10 am and 2 pm the arrival rate steadily increases from 5 per hour at 10 am to 25 per hour
at 2 pm. From 2 pm to 8 pm the arrival rate steadily decreases from 25 per hour at 2 pm
to 4 per hour at 8 pm. Between 8 pm and midnight the arrival rate is 3 an hour and from
midnight to 8 am the arrival rate is 1 per hour. The amounts of money withdrawn by the
customers are independent and identically distributed random variables with a mean of $100
and a standard deviation of $125.
(a) What is the probability distribution of the number of customers withdrawing money
during a 24-hour period?
(b) Calculate an approximation to the probability that the total withdrawal during 24 hours
is more than $25 000.
1.24 Parking-fee dodgers enter the parking lot of the University of Amsterdam according to
a Poisson process with rate λ. The parking lot has ample capacity. Each fee dodger parks
his/her car during an Erlang (2, µ) distributed time. It is university policy to inspect the
parking lot every T time units, with T fixed. Each newly arrived fee dodger is fined. What
is the probability distribution of the number of fee dodgers who are fined at an inspection?
1.25 Suppose customers arrive according to a non-stationary Poisson process with arrival rate
function λ(t). Any newly arriving customer is marked as a type k customer with probability
p k for k = 1, . . . , L, independently of the other customers. Prove that the customers of