A gamma [Gamma(α, θ)] variable has pdf
f(x; α, θ) =
(x⁄θ) α−1
e−x ⁄ θ ⁄
[Γ(α) θ], where
μ = α.θ and
σ² = α.θ². Also,
θ = σ² ⁄ μ and
α = (μ ⁄ σ)². Some
properties:
• If
X ~ Gamma(α, θ), then
X ⁄θ ~ Gamma(α, 1)
• Gamma(1, θ) = Exponential(θ)
• c X ~ Gamma(α, c θ) for
c>0
• For (independent) Xi ~
Exponential(θ),
Y = Σi=1, N
Xi : Y ~ Gamma(N, θ)
• Gamma(ν⁄2, 2) =
χ²(ν)
• For integer α = k,
Gamma(k, θ) = Erlang(k, θ)
• For large α, Gamma(α, θ)
→ Gauss(μ=α θ,
σ=√α θ)
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