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:

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, the mean and variance of X are E(X) = κβ and Var(X) = κβ2. When β = 1, the distribution is called a standard gamma distribution with parameter κ. We use the notation G ∼ gamma(κ) to denote that G follows a standard gamma distribution with parameter κ. The moments of G are

5.55

Weibull Distribution

A random variable X has a Weibull distribution with parameters α and β (α > 0, β > 0) if its pdf is given by

where β and α are the scale and shape parameters of the distribution. The mean and variance of X are

and the CDF of X is

When α = 1, the Weibull distribution reduces to an exponential distribution.

Define Y = X/[βΓ(1 + 1/α)]. We have E(Y) = 1 and the pdf of Y is

5.56

where the scale parameter β disappears due to standardization. The CDF of the standardized Weibull distribution is

and we have E(Y) = 1 and Var(Y) = Γ(1 + 2/α)/[Γ(1 + 1/α)]2 − 1. For a duration model with Weibull innovations, the pdf in Eq. (5.56) is used in the maximum-likelihood estimation.

Generalized Gamma Distribution

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

where β is a scale parameter, and α and κ are shape parameters. This distribution can be written as

where G is a standard gamma random variable with parameter κ. The pdf of X can be obtained from that of G by the technique of changing variables. Similarly, the moments of X can be obtained from that of G in Eq. (5.55) by

When κ = 1, the generalized gamma distribution reduces to that of a Weibull distribution. Thus, the exponential and Weibull distributions are special cases of the generalized gamma distribution.

The expectation of a generalized gamma distribution is E(X) = βΓ(κ + 1/α)/Γ(κ). In duration models, we need a distribution with unit expectation. Therefore, defining a random variable Y = λX/β, where λ = Γ(κ)/Γ(κ + 1/α), we have E(Y) = 1 and the pdf of Y is

5.57

where again the scale parameter β disappears and λ = Γ(κ)/Γ(κ + 1/α).