1.3.6.6.11. Gamma Distribution

1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.6. Probability Distributions
1.3.6.6. Gallery of Distributions

1.3.6.6.11. Gamma Distribution

Probability Density Function

The general formula for the probability density function of the gamma distribution is

f(x) = ((x-mu)/beta)*gamma**(-1)*EXP(-(x-mu)/beta)/
(beta*GAMMA(gamma)) x >= mu; gamma, beta > 0

where gamma is the shape parameter, is the location parameter, beta is the scale parameter, and GAMMA is the gamma function which has the formula

GAMMA(a) = INTEGRAL[t**(a-1)*EXP(-t)dt] where the
integration is from 0 to infinity

The case where = 0 and beta = 1 is called the standard gamma distribution. The equation for the standard gamma distribution reduces to

f(x) = x**(gamma-1)*EXP(-x)/GAMMA(gamma) for x >= 0

Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function.

The following is the plot of the gamma probability density function.

plot of the gamma probability density function

Cumulative Distribution Function

The formula for the cumulative distribution function of the gamma distribution is

F(x) = GAMMA(gamma,x)/GAMMA(gamma) for x >= 0

where GAMMA is the gamma function defined above and GAMMA(a,x) is the incomplete gamma function. The incomplete gamma function has the formula

GAMMA(a,x) = INTEGRAL[t**(a-1)*EXP(-t)dt] where the
integration is from 0 to x

The following is the plot of the gamma cumulative distribution function with the same values of gamma as the pdf plots above.

plot of the gamma cumulative distribution function

Percent Point Function

The formula for the percent point function of the gamma distribution does not exist in a simple closed form. It is computed numerically.

The following is the plot of the gamma percent point function with the same values of gamma as the pdf plots above.

plot of the gamma percent point function

Hazard Function

The formula for the hazard function of the gamma distribution is

h(x) = x**(gamma-1)*EXP(-x)/(GAMMA(gamma) - GAMMA(gamma,x))
for x >= 0

The following is the plot of the gamma hazard function with the same values of gamma as the pdf plots above.

plot of the gamma hazard function

Cumulative Hazard Function

The formula for the cumulative hazard function of the gamma distribution is

H(x) = -LOG[1 - GAMMA(gamma,x)/GAMMA(gamma)] x >= 0; gamma > 0

where GAMMA is the gamma function defined above and GAMMA(a,x) is the incomplete gamma function defined above.

The following is the plot of the gamma cumulative hazard function with the same values of gamma as the pdf plots above.

plot of the gamma cumulative hazard function

Survival Function

The formula for the survival function of the gamma distribution is

S(x) = 1 - GAMMA(gamma,x)/GAMMA(gamma) x >= 0; gamma > 0

where GAMMA is the gamma function defined above and GAMMA(a,x) is the incomplete gamma function defined above.

The following is the plot of the gamma survival function with the same values of gamma as the pdf plots above.

plot of the gamma survival function

Inverse Survival Function

The gamma inverse survival function does not exist in simple closed form. It is computed numberically.

The following is the plot of the gamma inverse survival function with the same values of gamma as the pdf plots above.

plot of the gamma inverse survival function

Common Statistics

The formulas below are with the location parameter equal to zero and the scale parameter equal to one.

Mean gamma

Mode gamma - 1

Range Zero to positive infinity.

Standard Deviation SQRT(gamma)

Skewness 2/SQRT(gamma)

Kurtosis 3 + 6/gamma

Coefficient of Variation 1/SQRT(gamma)

Parameter Estimation

The method of moments estimators of the gamma distribution are

gammahat = (xbar/s)**2

betahat = s**2/xbar

where xbar and s are the sample mean and standard deviation, respectively.

The equations for the maximum likelihood estimation of the shape and scale parameters are given in Chapter 18 of Evans, Hastings, and Peacock and Chapter 17 of Johnson, Kotz, and Balakrishnan. These equations need to be solved numerically; this is typically accomplished by using statistical software packages.

Software

Some general purpose statistical software programs, including Dataplot, support at least some of the probability functions for the gamma distribution.