Ralph Vince - Portfolio Mathematics

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

Probability Distributions 93

Suppose we have two independent random variables. The first of these,

Z, is standard normal (mean of 0 and variance of 1). The second of these,

which we call J, is Chi-Square distributed with V degrees of freedom. We

can now say that the variable T, equal to Z/(J/V), is distributed according

to the Student’s Distribution. We can also say that the variable T will follow

the Student’s Distribution with N – 1 degrees of freedom if:

T=

√

N∗(X−U)/S

where: X=A sample mean.

S=A sample standard deviation.

N=The size of a sample.

U=The population mean.

The probability density function for the Student’s Distribution, N′(X),

is given as:

N′(X)=

GAM((V+1)/2)

√

V∗P∗GAM(V/2)

∗(1+X^2 /V)−(V+1)/^2 (2.56)

where: P=pi, or 3.1415926536.

V=The degrees of freedom.

GAM( )=The standard gamma function.

The mathematics of the Student’s Distribution are related to the in-

complete beta function. Since we aren’t going to plunge into functions of

mathematical physics such as the incomplete beta function, we will leave

the Student’s Distribution at this point. Before we do, however, you still

need to know how to calculate probabilities associated with the Student’s

Distribution for a given number of standard units (Z score) and degrees of

freedom. As the following snippet of java code to discern the probabilities.

You’ll note that as the degrees of freedom variable, DEGFDM, approaches

infinity, the values returned, the probabilities, converge to the Normal as

given by Equation (2.22):

public void StudentsT2TailProbs(double zScore, int degreesOfFreedom){

double confidenceLevel = 1.0;

double st = Math.abs(zScore);

double r8 = Math.atan(st/Math.sqrt((double)degreesOfFreedom));

double rc8 = Math.cos(r8);

double x8 = 1.0;