# 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

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%.

### 10 Responses to Gamma Distribution

1. Winile says:

This was extremely helpful thank you very much!!

2. Elsa says:

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

• Charles says:

Elsa,
Yes, the Excel notation is the opposite of others.
Charles

3. 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

4. Dr buenas noches, muchas gracias por su página.
Dr,Real Statistics cuenta con la regresión de Poisson?
Gracias

Dr Good evening, thank you very much for your page.
Dr, Real Statistics has the Poisson regression?
Thank you

• Charles says:

Gerardo,
Real Statistics doesn-t yet support Poisson regression, but I do plan to add this shortly.
Charles

• DR. Muchas gracias, sin embargo pensé que por modelo log-lineal, podría aplicarla.

Gracias

DR. Many thanks, however I thought that by log-linear model, I could apply it.

Thank you

5. Niraj Kaji says:

If I am seeking to create a gamma distribution in Excel, how would I accomplish this? For example, I am seeing to create a set of sample data for wages (which often has a gamma type distribution). I know what the 10th, 25th, 50th, 75th and 90th percentile wages are…