Ralph Vince - Portfolio Mathematics

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

Probability Distributions 87

FIGURE 2.30 Cumulative probability functions for the Exponential Distribution

(A=1)

Again A is the single parametric input, equal to 1/L in the Poisson Distribu-

tion, and must be greater than 0.

Another interesting quality about the Exponential Distribution is that it

has what is known as the “forgetfulness property.” In terms of a telephone

switchboard, this property states that the probability of a call in a given

time interval is not affected by the fact that no calls may have taken place

in the preceding interval(s).

The Chi-Square Distribution

A distribution that is used extensively in goodness-of-fit testing is theChi-

Square Distribution(pronounced ki square, from the Greek letter X (chi)

and hence often represented as the X^2 distribution).

Assume that K is a standard normal random variable (i.e., it has mean

0 and variance 1). If we say that K equals the square root of J (J=K^2 ) , then

we know that K will be a continuous random variable. However, we know

that K will not be less than zero, so its density function will differ from the

Normal. The Chi-Square Distribution gives us the density function of K:

N′(K)=KV/^2 −^1 ∗EXP(−V/2)/ 2 V/^2 ∗GAM(V/2) (2.51)