ABSTRACT
The gamma distribution can be viewed as a generalization of the exponential distribution with mean 1/λ, λ > 0. An exponential random variable with mean 1/λ represents the waiting time until the first event to occur, where events are generated by a Poisson process with mean λ, while the gamma random variable X represents the waiting time until the ath event to occur. Therefore,
Yi,
where Y1, . . . , Yn are independent exponential random variables with mean 1/λ. The probability density function of X is given by
f(x|a, b) = 1 Γ(a)ba
e−x/bxa−1, x > 0, a > 0, b > 0, (15.1.1)
where b = 1/λ. The distribution defined by (15.1.1) is called the gamma distribution with shape parameter a and the scale parameter b. It should be noted that (15.1.1) is a valid probability density function for any a > 0 and b > 0. The gamma distribution with a positive integer shape parameter a is called the Erlang Distribution. If a is a positive integer, then
F (x|a, b) = P (waiting time until the ath event is at most x units of time) = P (observing at least a events in x units of time when the
mean waiting time per event is b)
when number of events is x/b)
k!
