Ralph Vince - Portfolio Mathematics

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

Probability Distributions 71

The Uniform Distribution is acontinuousdistribution. The probability

density function, N′(X), is described as:

N′(X)= 1 /(B−A) for A<=X<=B (2.27)

else

N′(X)= 0

where: B=The rightmost limit of the interval AB.

A=The leftmost limit of the interval AB.

The cumulative density of the Uniform is given by:

N(X)=0forX<A (2.28)

else

N(X)=(X−A)/(B−A) for A<=X<=B

else

N(X)=1forX>B

where: B=The rightmost limit of the interval AB.

A=The leftmost limit of the interval AB.

Figures 2.16 and 2.17 illustrate the probability density and cumulative

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

Other qualities of the Uniform Distribution are:

Mean=(A+B)/ 2 (2.29)

Variance=(B−A)^2 / 12 (2.30)

where: B=The rightmost limit of the interval AB.

A=The leftmost limit of the interval AB.

The Bernoulli Distribution

Another simple, common distribution is theBernoulli Distribution.This

is the distribution when the random variable can have only two possible

values. Examples of this are heads and tails, defective and nondefective