Page 41 - Computational Statistics Handbook with MATLAB

Chapter 2: Probability Concepts 27

The expected value and variance of a Poisson random variable are both λ,
thus,

EX[] = λ ,

and

VX() = λ .

The Poisson distribution can be used in many applications. Examples of sit-
uations where a discrete random variable might follow a Poisson distribution
are:

• the number of typographical errors on a page,
• the number of vacancies in a company during a month, or
• the number of defects in a length of wire.

The Poisson distribution is often used to approximate the binomial. When
n is large and p is small (so np is moderate), then the number of successes
occurring can be approximated by the Poisson random variable with param-
eter λ = np .
The Poisson distribution is also appropriate for some applications where
events occur at points in time or space. We see it used in this context in Chap-
ter 12, where we look at modeling spatial point patterns. Some other exam-
ples include the arrival of jobs at a business, the arrival of aircraft on a
runway, and the breakdown of machines at a manufacturing plant. The num-
ber of events in these applications can be described by a Poisson process.
Let Nt() , t ≥ 0 , represent the number of events that occur in the time inter-
val 0 t,[ ] . For each interval 0 t,[ ] , Nt() is a random variable that can take on
values 01 2 …,, , . If the following conditions are satisfied, then the counting
λ
process {Nt() , t ≥ 0 } is said to be a Poisson process with mean rate [Ross,
2000]:

1. N 0() = . 0
2. The process has independent increments.
3. The number Nt() of events in an interval of length t follows a
Poisson distribution with mean λt . Thus, for s ≥ 0 , t ≥ , 0

λt λt( ) k
(
–
,,
(
PN t + s) – Ns() = k) = e ------------; k = 01 … . (2.26)
k!
From the third condition, we know that the process has stationary incre-
ments. This means that the distribution of the number of events in an interval
depends only on the length of the interval and not on the starting point. The

© 2002 by Chapman & Hall/CRC