Analysis of Financial Time Series, Third Edition

Appendix A: Review of Some Probability Distributions

Exponential Distribution

A random variable X has an exponential distribution with parameter β > 0 if its probability density function (pdf) is given by

Denoting such a distribution by X ∼ exp(β), we have E(X) = β and Var(X) = β2. The cumulative distribution function (CDF) of X is

When β = 1, X is said to have a standard exponential distribution.

Gamma Function

For κ > 0, the gamma function Γ(κ) is defined by

The most important properties of the gamma function are:

1. For any κ > 1, Γ(κ) = (κ − 1)Γ(κ − 1).

2. For any positive integer m, Γ(m) = (m − 1)!.

3. .

The integration

is an incomplete gamma function. Its values have been tabulated in the literature. Computer programs are now available to evaluate the incomplete gamma function.

Gamma Distribution

A random variable X has a gamma distribution with parameter κ and β (κ > 0, β > 0) if its pdf is given by

By changing variable y = x/β, one can easily obtain the moments of X:

In particular, ...