Gamma distribution STAT 587 (Engineering) Iowa State University September 17, 2020

Gamma distribution Probability density function Gamma distribution The random variable X has a gamma distribution with shape parameter α > 0 and rate parameter λ > 0 if its probability density function is λ α Γ( α ) x α − 1 e − λx I( x > 0) p ( x | α, λ ) = where Γ( α ) is the gamma function, � ∞ x α − 1 e − x dx. Γ( α ) = 0 We write X ∼ Ga ( α, λ ) .

Gamma distribution Probability density function - graphically Gamma probability density function Gamma random variables rate = 0.5 rate = 1 rate = 2 3 shape = 0.5 2 Probablity density function, f(x) 1 0 2.0 1.5 shape = 1 1.0 0.5 0.0 0.6 shape = 2 0.4 0.2 0.0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 x

Gamma distribution Mean and variance Gamma mean and variance If X ∼ Ga ( α, λ ) , then � ∞ x λ α Γ( α ) x α − 1 e − λx dx = · · · = α E [ X ] = λ 0 and � ∞ � 2 λ α x − α Γ( α ) x α − 1 e − λx dx = · · · = α � V ar [ X ] = λ 2 . λ 0

Gamma distribution Cumulative distribution function Gamma cumulative distribution function If X ∼ Ga ( α, λ ) , then its cumulative distribution function is � x λ α Γ( α ) t α − 1 e − λt dt = · · · = γ ( α, βx ) F ( x ) = Γ( α ) 0 where γ ( α, βx ) is the incomplete gamma function, i.e. � βx t α − 1 e − t dt. γ ( α, βx ) = 0

Gamma distribution Cumulative distribution function - graphically Gamma cumulative distribution function - graphically Gamma random variables rate = 0.5 rate = 1 rate = 2 1.00 shape = 0.5 0.75 0.50 Cumulative distribution function, F(x) 0.25 0.00 1.00 0.75 shape = 1 0.50 0.25 0.00 1.00 0.75 shape = 2 0.50 0.25 0.00 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 x

Gamma distribution Relationship to exponential distribution Relationship to exponential distribution iid If X i ∼ Exp ( λ ) , then n � Y = X i ∼ Ga ( n, λ ) . i =1 Thus, Ga (1 , λ ) d = Exp ( λ ) .

Gamma distribution Parameterization by the scale Parameterization by the scale A common alternative parameterization of the Gamma distribution uses the scale θ = 1 λ . In this parameterization, we have 1 Γ( α ) θ α x α − 1 e − x/θ I( x > 0) f ( x ) = and V ar [ X ] = αθ 2 . E [ X ] = αθ and

Gamma distribution Summary Summary Gamma random variable X ∼ Ga ( α, λ ) , α, λ > 0 Γ( α ) x α − 1 e − λx , x > 0 λ α f ( x ) = E [ X ] = α λ V ar [ X ] = α λ 2

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