Statistics – Gamma Distribution
The gamma distribution represents continuous probability distributions of two-parameter family. Gamma distributions are devised with generally three kind of parameter combinations.
- A shape parameter kk and a scale parameter θθ.
- A shape parameter α=kα=k and an inverse scale parameter β=1θβ=1θ, called as rate parameter.
- A shape parameter kk and a mean parameter μ=kβμ=kβ.
Each parameter is a positive real numbers. The gamma distribution is the maximum entropy probability distribution driven by following criteria.
Formula
Where −
- XX = Random variable.
- ψψ = digamma function.
Characterization using shape αα and rate ββ
Probability density function
Probability density function of Gamma distribution is given as:
Formula
Where −
- αα = location parameter.
- ββ = scale parameter.
- xx = random variable.
Cumulative distribution function
Cumulative distribution function of Gamma distribution is given as:
Formula
Where −
- αα = location parameter.
- ββ = scale parameter.
- xx = random variable.
- γ(α,βx)γ(α,βx) = lower incomplete gamma function.
Characterization using shape kk and scale θθ
Probability density function
Probability density function of Gamma distribution is given as:
Formula
