The Gamma Distribution can be used to represent the waiting time until the nth occurrence of an event, assuming a constant occurrence rate per unit time. Some examples are the time to the arrival of a fixed number of vehicles (to a given location) or the time to failure of a system designed to accept a fixed number of overloads before failing.
The Gamma Distribution is defined for all positive values of x. It includes both a power of x and a term that declines exponentially. Its parameters are a and b, both of which must be positive. The distribution is:
where x>0, a>0, and b>0. The gamma function, , is a generalization of the factorial function. Factorials are only defined for integers, but the gamma function can have any number for an argument. If n is an integer, = (n-1)!.
The equations for the mean and variance are:
The equations for the parameters are:
The mean must be positive.
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