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
E[X]=kθ=αβ>0 and is fixed.E[ln(X)]=ψ(k)+ln(θ)=ψ(α)−ln(β) and is fixed.E[X]=kθ=αβ>0 and is fixed.E[ln(X)]=ψ(k)+ln(θ)=ψ(α)−ln(β) and is fixed.
Where −
· XX = Random variable.
· ψψ = digamma function.
Characterization using shape ααand rateββ
Probability density function
Probability density function of Gamma distribution is given as:
Formula
f(x;α,β)=βαxα−1e−xβΓ(α) where x≥0 and α,β>0f(x;α,β)=βαxα−1e−xβΓ(α) where x≥0 and α,β>0
Where −
· αα = location parameter.
· ββ = scale parameter.
· xx = random variable.
Cumulative distribution function
Cumulative distribution function of Gamma distribution is given as:
Formula
F(x;α,β)=∫x0f(u;α,β)du=γ(α,βx)Γ(α)F(x;α,β)=∫0xf(u;α,β)du=γ(α,βx)Γ(α)
Where −
· αα = location parameter.
· ββ = scale parameter.
· xx = random variable.
· γ(α,βx)γ(α,βx) = lower incomplete gamma function.
Characterization using shape kkandscaleθθ
Probability density function
Probability density function of Gamma distribution is given as:
Formula
f(x;k,θ)=xk−1e−xθθkΓ(k) where x>0 and k,θ>0f(x;k,θ)=xk−1e−xθθkΓ(k) where x>0 and k,θ>0
Where −
· kk = shape parameter.
· θθ = scale parameter.
· xx = random variable.
· Γ(k)Γ(k) = gamma function evaluated at k.
Cumulative distribution function
Cumulative distribution function of Gamma distribution is given as:
Formula
F(x;k,θ)=∫x0f(u;k,θ)du=γ(k,xθ)Γ(k)F(x;k,θ)=∫0xf(u;k,θ)du=γ(k,xθ)Γ(k)
Where −
· kk = shape parameter.
· θθ = scale parameter.
· xx = random variable.
· γ(k,xθ)γ(k,xθ) = lower incomplete gamma function.
