Gamma Distribution: Shape, Rate, and Connection to Poisson

Definition and Parameters

If X ~ Gamma(alpha, lambda), the PDF is f(x) = (lambda^alpha / Gamma(alpha)) * x^(alpha-1) * e^(-lambda*x) for x > 0. Here alpha is the shape parameter and lambda is the rate parameter. The mean is alpha/lambda and the variance is alpha/lambda^2. The gamma function Gamma(alpha) = integral of t^(alpha-1) * e^(-t) dt from 0 to infinity, with Gamma(n) = (n-1)! for positive integers.

Some sources use the scale parameterization with theta = 1/lambda, giving mean alpha*theta and variance alpha*theta^2. Always check which form a problem uses.

Special Cases

The exponential distribution is Gamma(1, lambda). The chi-squared distribution with k degrees of freedom is Gamma(k/2, 1/2). When alpha is a positive integer, the gamma distribution is also called the Erlang distribution. These connections are tested on Exam P.

The sum of n independent Exponential(lambda) random variables has a Gamma(n, lambda) distribution. More generally, if X ~ Gamma(alpha1, lambda) and Y ~ Gamma(alpha2, lambda) are independent (same rate), then X + Y ~ Gamma(alpha1 + alpha2, lambda).

Connection to the Poisson Process

In a Poisson process with rate lambda, the waiting time until the alpha-th event follows a Gamma(alpha, lambda) distribution. This connection is useful on Exam P: the probability that the alpha-th event occurs after time t equals P(Gamma > t) = P(Poisson count in [0,t] < alpha). This relationship lets you convert between gamma probabilities and Poisson sums, which can simplify calculations when alpha is a small integer.