ch02 JWBK035-Vince February 12, 2007 6:50 Char Count= 0
82 HANDBOOK OF PORTFOLIO MATHEMATICS
The main difference between the Poisson and the Binomial distribu-
tions is that the Binomial is not appropriate for events that can occur more
than once within a given time frame. Such an example might be the prob-
ability of an automobile accident over the next 6 months. In the Binomial
we would be working with two distinct cases: Either an accident occurs,
with probability P, or it does not, with probability Q (i.e., 1 – P). However,
in the Poisson Distribution we can also account for the fact that more than
one accident can occur in this time period.
The probability density function of the Poisson, N′(X), is given by:N′(X)=(LX∗EXP(−L))/X! (2.43)
where: L=The parameter of the distribution.
EXP( )=The exponential function.Note that X must take discrete values.
Suppose that calls to a switchboard average four calls per minute
(L=4). The probability of three calls (X=3) arriving in the next minute
is:N′(3)=(4^3 ∗EXP(−4))/3!
=(64∗EXP(−4))/(3∗2)
=(64∗.01831564)/ 6
= 1. 17220096 / 6
=. 1953668267
So we can say there is about a 19.5% chance of getting three calls in
the next minute. Note that this is not cumulative—that is, this is not the
probability of getting three calls or fewer, it is the probability of getting
exactly three calls. If we wanted to know the probability of getting three
calls or fewer, we would have had to use the N(3) formula [which is given
in (2.46)].
Other properties of the Poisson Distribution are:Mean=L (2.44)
Variance=L (2.45)where: L=The parameter of the distribution.