The gamma distribution has the same relationship to the Poisson distribution that the negative binomial distribution has to the binomial distribution. We aren’t going to study the gamma distribution directly, but it is related to the exponential distribution and especially to the chi-square distribution which will receive a lot more attention in this website.
Definition 1: The gamma distribution has probability density function (pdf) given by
Excel Function: Excel provides the following functions:
GAMMADIST(x, α, β, cum) where α, β are the parameters in Definition 1 and cum = TRUE or FALSE
GAMMADIST(x, α, β, FALSE) = f(x) where f is the pdf as defined above
GAMMADIST(x, α, β, TRUE) = F(x) where F is the cumulative distribution function corresponding to above
GAMMAINV(p, α, β) = x such that GAMMADIST(x, α, β, TRUE) = p. Thus GAMMAINV is the inverse of the cumulative distribution version of GAMMADIST.
Excel 2010/2013 also provide the following additional functions: GAMMA.DIST which is equivalent to GAMMADIST and GAMMA.INV which is equivalent to GAMMAINV.
Example 1: Suppose that sending a money order is a random event and that at a particular shop someone sends a money order on average every 15 minutes. What is the probability that the shop sends a total of 10 money orders in less than 3 hours?
For this problem λ = 4 money orders per hour. Let x = the time to send 10 money orders and let F(x) be the cumulative gamma distribution function with α = k = 10 and β = 1/λ = .25. Thus
P(x<3) = F(3) = GAMMADIST(3, 10, .25, TRUE) = .7586
and so the probability is 75.86%.