Ralph Vince - Portfolio Mathematics

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

78 HANDBOOK OF PORTFOLIO MATHEMATICS

. 2499 /. 00001111 =N− 1

. 2499 /. 00001111 + 1 =N

22 , 491 + 1 =N

N= 22 , 492

Thus, we need to witness a 51% win rate over 22,492 trials to be 99.865%

certain that we will see at least 51% wins.

The Geometric Distribution

Like the Binomial, theGeometric Distribution, also a discrete distribution,

occurs as a result of N independent Bernoulli trials. The Geometric Distri-

bution measures the number of trials before the first success (or failure).

The probability density function, N′(X), is:

N′(X)=Q(X−1)∗P (2.39)

where: P=The probability of success for a given trial.

Q=The probability of failure for a given trial.

In other words, N′(X) here measures the number of trials until the first

success. The cumulative density function for the Geometric is therefore:

N(X)=

∑X

J= 1

Q(J−1)∗P (2.40)

where: P=The probability of success for a given trial.

Q=The probability of failure for a given trial.

Figures 2.22 and 2.23 illustrate the probability density and cumulative

probability ability (i.e., cdf), respectively, of the Geometric Distribution.

Other properties of the Geometric are:

Mean= 1 /P (2.41)

Variance=Q/P^2 (2.42)

where: P=The probability of success for a given trial.

Q=The probability of failure for a given trial.