Gamma distribution: characteristics, applications

Definition

A continuous random variable X has a gamma distribution with shape parameter α > 0 and scale parameter β > 0 if its pdf is

f(x) = x^(α−1) e^(−x/β) / (βᵅ Γ(α)), x > 0, and 0 elsewhere,

where Γ(α) = ∫₀^∞ t^(α−1) e^(−t) dt is the gamma function (Γ(n) = (n − 1)! for integer n, Γ(1/2) = √π).

Chief characteristics

1. Defined for x > 0 only; it is positively skewed.
2. Mean = αβ, variance = αβ², so standard deviation = β√α.
3. Mode = (α − 1)β for α ≥ 1.
4. The shape depends on α: for α = 1 it reduces to the exponential distribution with mean β; for α = ν/2 and β = 2 it is the chi-square distribution with ν degrees of freedom.
5. As α grows large, it tends to a normal shape.
6. When α is a positive integer it models the waiting time until the α-th event of a Poisson process (Erlang distribution).
7. The sum of independent gamma variables with the same β is again gamma, with the α's added.
8. The mgf is (1 − βt)^(−α), t < 1/β.

Applications

• Waiting time or time to failure until the α-th event (queues, reliability of systems with standby units).
• Service times and lifetimes of components.
• Daily rainfall, river flow and flood levels (hydrology).
• Daily electricity or water consumption.
• Insurance claim sizes and income distribution.
• Through the chi-square case, it underlies tests of variance and goodness of fit.