Statistics – Gamma Distribution

Statistics-Gamma-Distribution

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.

Gamma Distribution

Each parameter is a positive real numbers. The gamma distribution is the maximum entropy probability distribution driven by following criteria.

Formula

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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

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Where −

  • αα = location parameter.
  • ββ = scale parameter.
  • xx = random variable.

Cumulative distribution function

Cumulative distribution function of Gamma distribution is given as:

Formula

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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

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