Gamma Distribution

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

Gamma Distribution PDF

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%.

5 Responses to Gamma Distribution

  1. Anthony Grieb says:

    Why is so little emphasis placed on the Gamma distribution? Does it not play much of a role in statistics?

    It is important for population modeling and queuing theory. But I suppose these are more of mathematical models than they are statistics.

    Empirically speaking

    k=μ * lambda

    When k=1 this is the exponential distribution.

    k>=20 is essentially normal

    And k<1 suggests a branching process following a Borel distribution or a more complicated distribution that takes lifespan and age into account.

    • Charles says:

      Anthony,
      Thanks for your comments. The gamma distribution does play a role in statistics (esp. the exponential distribution), but not so much in the topics we have covered thus far. When I was working in a company where queueing models were used, this distribution was very important.
      Charles

  2. Elsa says:

    So the Excel notation is alpha_Excel = shape = alpha_Wikipedia and beta_Excel = scale = 1/beta_Wikipedia?

  3. Winile says:

    This was extremely helpful thank you very much!!

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