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Gamma distribution

Gamma Distribution

Probability Density FunctionThe general formula for the probability density function of the gamma distribution isf(x) = ((x-mu)/beta)*gamma**(-1)*EXP(-(x-mu)/beta)/ (beta*GAMMA(gamma)) x >= mu; gamma, beta > 0
where gamma is the shape parameter, mu 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 mu = 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 FunctionThe formula for the cumulative distribution function of the gamma distribution isF(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 FunctionThe 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 FunctionThe formula for the hazard function of the gamma distribution ish(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 FunctionThe formula for the cumulative hazard function of the gamma distribution isH(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 FunctionThe formula for the survival function of the gamma distribution isS(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 FunctionThe 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 StatisticsThe formulas below are with the location parameter equal to zero and the scale parameter equal to one.
Meangamma
Modegamma - 1
RangeZero to positive infinity.
Standard DeviationSQRT(gamma)
Skewness2/SQRT(gamma)
Kurtosis3 + 6/gamma
Coefficient of Variation1/SQRT(gamma)
Parameter EstimationThe method of moments estimators of the gamma distribution aregammahat = (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.
SoftwareSome general purpose statistical software programs support at least some of the probability functions for the gamma distribution.

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