Ralph Vince - Portfolio Mathematics

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

Probability Distributions 75

FIGURE 2.21 Cumulative probability functions for the Binomial Distribution

(N=5, P=.5)

where: N=The number of trials.

P=The probability of a success on a single trial.

Q= 1 −P.

As N becomes large, the Binomial tends to the Normal Distribution,

with the Normal being the limiting form of the Binomial. Generally, if N * P

and N * Q are both greater than 5, you could use the Normal in lieu of the

Binomial as an approximation.

The Binomial Distribution is often used to statistically validate a gam-

bling system. An example will illustrate. Suppose we have a gambling sys-

tem that has won 51% of the time. We want to determine what the winning

percentage would be if it performs in the future at a level of 3 standard

deviations worse. Thus, the variable of interest here, X, is equal to .51, the

probability of a winning trade. The variable of interest need not always be

for the probability of a win. It can be the probability of an event being in

one of two mutually exclusive groups. We can now perform the first neces-

sary equation in the test:

L=P−Z∗

√

(P∗(P−1))/(N−1) (2.38)