Statistics/Distributions/Gamma

Gamma Distribution Gamma density 325px|Probability density plots of gamma distributions 325px|Cumulative distribution plots of gamma distributions shape scale (see digamma function) No simple closed form (see trigamma function ) The Gamma distribution is very important for technical reasons, since it is the parent of the exponential distribution and can explain many other distributions. The probability distribution function is: Where is the Gamma function. The cumulative distribution function cannot be found unless p=1, in which case the Gamma distribution becomes the exponential distribution. The Gamma distribution of the stochastic variable X is denoted as . Alternatively, the gamma distribution can be parameterized in terms of a shape parameter and an inverse scale parameter , called a rate parameter: where the constant can be calculated setting the integral of the density function as 1: following: and, with change of variable : following: Probability Density Function We first check that the total integral of the probability density function is 1. Now we let y=x/a which means that dy=dx/a Mean Now we let y=x/a which means that dy=dx/a. We now use the fact that Variance We first calculate E[X^2] Now we let y=x/a which means that dy=dx/a. Now we use calculate the variance External links Interactive Gamma Distribution Web Applet (Java)