Gamma distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ gamma(α,β) is used to indicate that the random variable X has the gamma distribution. A gamma random variable X with positive scale parameter α and positive shape parameter β has probability density function

f(x) = \frac{x^{β−1} e^{−x/α}}{α^{β}Γ(β)} x > 0.

The gamma distribution can be used to model service times, lifetimes of objects, and repair times. The gamma distribution has an exponential right-hand tail. The probability density function with several parameter combinations is illustrated below.

PSfrag replacements α = 2, β = 1

α = 1, β = 2
α = 1, β = 5
x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = \frac{Γ(β,x/α)}{Γ(β)} x > 0,

where

Γ(s,x) = ∫_{0}^{x}t^{s−1} e^{−t}dt

for s > 0 and x > 0 is the incomplete gamma function and

Γ(s) = ∫_{0}^{∞} t^{s−1} e^{−t}dt

for s > 0 is the gamma function. The survivor function on the support of X is

S(x) = P(X ≥ x) = 1 − \frac{Γ(β,x/α)}{Γ(β)} x > 0.

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \frac{x^{β−1} e^{−x/α}}{(Γ(β)− Γ (β,x/α)) α^{β}Γ(β)} x > 0.

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = −ln( 1 − \frac{Γ(β,x/α)}{Γ(β)} ) x > 0.

There is no closed-form expression for the inverse distribution function. The moment generating function of X is

M(t) = E [e^{tX}] = (1 − αt)^{−β} t < \frac{1}{α}.

The characteristic function of X is

ϕ(t) = E [e^{itX}] = (1 − αit)^{−β} t < \frac{1}{α}.

The population mean, variance, skewness, and kurtosis of X are

E[X] = αβ  V [X] = α^{2}β  E [( \frac{X − μ}{σ} ) ^{3}] = \frac{2}{\sqrt{β}}  E [( \frac{X − μ}{σ} ) ^{4}] = 3+ \frac{6}{β}.

For X_{1}, X_{2}, ..., X_{n} mutually independent gamma(α,β) random variables, the method of moments for α and β are

α̂ = s^{2}/x̄

and

β̂ = (x̄/s)^{2}.

APPL verification: The APPL statements

assume(alpha ¿ 0); assume(beta ¿ 0); X := [[x -¿ x ˆ (beta - 1) * exp(-x / alpha) / (alpha ˆ beta * GAMMA(beta))],

[0, infinity], [”Continuous”, ”PDF”]]: Mean(X); Variance(X); Skewness(X); Kurtosis(X);

verify the population mean, variance, skewness, and kurtosis.