The gamma distribution is mathematically defined from the gamma function. This post gives a brief introduction to the beta function. The goal is to establish one property that is the basis for defining the beta distribution.
The Beta Function
For any positive constants and , the beta function is defined to be the following integral:
The beta function can be evaluated directly if the parameters and are not too large. For example, is the integral , which is . Evaluating in a case by case basis does not shed light on the beta function. Direct calculation can also be cumbersome (e.g. for large parameters that are integers) or challenging (e.g. for parameters and that are fractional). It turns out that the evaluation of the beta function is based on the gamma function.
Connection to the Gamma Function
The remainder of the post is to establish the following value of the beta function:
To start the proof of , let and be two independent random variables such that follows a gamma distribution with shape parameter and rate parameter and that follows a gamma distribution with shape parameter and rate parameter . It does not matter what is, as long as it is the rate parameter for both and . Then the sum has a gamma distribution with shape parameter and rate parameter . The following is the density function for .
The density function of can also be derived from the convolution formula using the density functions of and as follows:
See here for more information on how to use the convolution formula. The last step in is obtained by a change of variable in the integral from the step immediately above it by letting . The last step in must equal to . Setting the two equal would produce the equality in .
Note that if the function is normalized by the value , it would be a density function, which is the beta distribution. The following is the density function of the beta distribution.
The beta distribution is further examined in the next post.