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 and exponential distribution.

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

Also, Read

Gamma Distribution Formula

Gamma Distribution,

where p and x are a continuous random variable.

Gamma Distribution

Gamma Distribution Cumulative Distribution Function

Gamma Distribution

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

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:

Gamma Distribution

Gamma Distribution Variance

It can be shown as follows:

Gamma Distribution

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