Ralph Vince - Portfolio Mathematics

ch02 JWBK035-Vince February 12, 2007 6:50 Char Count= 0

Probability Distributions 83

FIGURE 2.25 Probability density functions for the Poisson Distribution

(L=.5)

In the Poisson Distribution, both the mean and the variance equal the

parameter L. Therefore, in our example case we can say that the mean

is four calls and the variance is four calls (or, the standard deviation is 2

calls—the square root of the variance, 4).

When this parameter, L, is small, the distribution is shaped like a

reversed J, and when L is large, the distribution is not dissimilar to

the Binomial. Actually, the Poisson is the limiting form of the Bi-

nomial as N approaches infinity and P approaches O. Figures 2.25

through 2.28 show the Poisson Distribution with parameter values of .5

and 4.5.

The cumulative density function of the Poisson, N(X), is given by:

N(X)=

∑X

J= 0

(LJ∗EXP(−L))/J! (2.46)

where:

L=The parameter of the distribution.

EXP( )=The exponential function.