Gamma Distribution

What is 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

Gamma Distribution Formula

where p and x are a continuous random variable.

Gamma Distribution Graph

The parameters of the gamma distribution define 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 below:

Gamma Distribution Cumulative Distribution Function

Gamma Distribution Properties

The properties of the gamma distribution are:

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