Introduction to Probability, Second Edition, 2nd Edition

8.4    Gamma

The Gamma distribution is a continuous distribution on the positive real line; it is a generalization of the Exponential distribution. While an Exponential r.v. represents the waiting time for the first success under conditions of memorylessness, we shall see that a Gamma r.v. represents the total waiting time for multiple successes.

Before writing down the PDF, we first introduce the gamma function, a very famous function in mathematics that extends the factorial function beyond the realm of nonnegative integers.

Definition 8.4.1 (Gamma function). The gamma function $\text{Γ}$ is defined by

$\text{Γ}\left(a\right)={\int }_0^{\infty }{x}^a{e}^{-x}\frac{dx}{x},$

for real numbers $a>0$.

We could also cancel an x and write the integrand as $x^{a-1}{e}^{-x}$, but it turns out to be convenient having the ...