Page 27 - A First Course In Stochastic Models

18 THE POISSON PROCESS AND RELATED PROCESSES

case caused by the pedestrian’s carelessness. In the preceding years such accidents
occurred on average 3.7 times per year. Is the clustering of accidents in the year
2000 exceptional? It is exceptional if seven or more fatal accidents occur during
the coming five months, but it is not exceptional when over a period of ten years
(say) seven or more accidents happen in some time window having a length of
five months. The above approximation gives the value 0.104 for the probability
that over a period of ten years there is some time window having a length of
five months in which seven or more fatal accidents occur. The exact value of the
probability is 0.106.

1.2 COMPOUND POISSON PROCESSES

A compound Poisson process generalizes the Poisson process by allowing jumps
that are not necessarily of unit magnitude.

Definition 1.2.1 A stochastic process {X(t), t ≥ 0} is said to be a compound
Poisson process if it can be represented by

N(t)

X(t) = D i , t ≥ 0,
i=1

where {N(t), t ≥ 0} is a Poisson process with rate λ, and D 1 , D 2 , . . . are inde-
pendent and identically distributed non-negative random variables that are also
independent of the process {N(t)}.

Compound Poisson processes arise in a variety of contexts. As an example,
consider an insurance company at which claims arrive according to a Poisson
process and the claim sizes are independent and identically distributed random
variables, which are also independent of the arrival process. Then the cumulative
amount claimed up to time t is a compound Poisson variable. Also, the compound
Poisson process has applications in inventory theory. Suppose customers asking
for a given product arrive according to a Poisson process. The demands of the
customers are independent and identically distributed random variables, which are
also independent of the arrival process. Then the cumulative demand up to time t
is a compound Poisson variable.
The mean and variance of the compound Poisson variable X(t) are given by

2
2
E[X(t)] = λtE(D 1 ) and σ [X(t)] = λtE(D ), t ≥ 0. (1.2.1)
1
This result follows from (A.9) and (A.10) in Appendix A and the fact that both
the mean and variance of the Poisson variable N(t) are equal to λt.