## - Gamma Distribution

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. Notice that X =

∑a i 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, (16.1)

where b = 1/λ. The distribution defined by (16.1) is called the gamma distribution with shape parameter a and the scale parameter b. It should be noted that (16.1) is a valid probability density function (pdf) for any a > 0 and b > 0. The cumulative distribution function is given by

F (x|a, b) = 1 Γ(a)ba

e−t/bta−1dt, x > 0, a > 0, b > 0.