Gamma Distribution

The gamma distribution term is mostly used as a distribution which is defined as two parameters - shape parameter and inverse scale parameter, having continuous probability distributions. It is related to the normal distribution,  exponential distribution, chi-squared distribution and Erlang distribution. ‘Γ’ denotes the gamma function.

Gamma distributions have two free parameters, named as alpha (α) and beta (β), where;

  • α = Shape parameter
  • β = Rate parameter (the reciprocal of the scale parameter)

It is characterized by mean µ=αβ and variance σ2=αβ2

The scale parameter β is used only to scale the distribution. This can be understood by remarking that wherever the random variable x appears in the probability density, then it is divided by β. Since the scale parameter provides the dimensional data, it is seldom useful to work with the “standard” gamma distribution, i.e., with β = 1.

Gamma Distribution Function

The gamma function is represented by Γ(y) which is an extended form of factorial function to complex numbers(real). So, if n∈{1,2,3,…}, then Γ(y)=(n-1)!

If α is a positive real number, then Γ(α) is defined as

  • Γ(α) = 0∫∞ ( ya-1e-y dy) , for α > 0.
  • If α = 1, Γ(1) =0∫∞ (e-y dy) = 1
  • If we change the variable to y = λz, we can use this definition for gamma distribution: Γ(α) = 0∫∞ ya-1 eλy dy where α, λ >0.

Gamma Distribution Formula

,

where p and x are a continuous random variable.

Gamma Distribution Graph

The paremeters of the gamma distribution defines the shape of the graph. Shape parameter α and rate parameter β are both greater than 1.

  • When α = 1, this becomes the exponential distribution
  • When β = 1 this becomes the standard gamma distribution

Gamma Distribution of Cumulative Distribution Function

The cumulative distribution function of a Gamma distribution is as shown above:

Gamma Distribution Properties

For any +ve real number α,

  • Γ(α) = 0∫∞ ( ya-1e-y dy) , for α > 0.
  • 0∫∞ ya-1 eλy dy = Γ(α)/λa, for λ >0.
  • Γ(α +1)=α Γ(α)
  • Γ(m)=(m-1)!, for m = 1,2,3 …;
  • Γ(½) = √π

Gamma Distribution Mean

There are two ways to determine the gamma distribution mean

  1. Directly
  2. Expanding the moment generation function

It is also known as the Expected value of Gamma Distribution.

Gamma Distribution Variance

It can be shown as follows:

So, Variance = E[x2] – [E(x2)], where p = (E(x)) (Mean and Variance p(p+1) – p2 = p

Gamma Distribution Example

Imagine you are solving difficult Maths theorems and you expect to solve one every 1/2 hour. Compute the probability that you will have to wait between 2 to 4 hours before you solve four of them.

One theorem every 1/2 hour means we would suppose to get θ = 1 / 0.5 = 2 theorem every hour on average. Using θ = 2 and k = 4, Now we can calculate it as follows: