Download Gamma distribution (from http://www.math.wm.edu/˜leemis/chart

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) =
x β−1 e−x/α
αβ 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.
f (x)
0.5
α = 2, β = 1
0.4
α = 1, β = 2
0.3
0.2
α = 1, β = 5
0.1
0.0
x
0
2
4
6
8
10
12
The cumulative distribution function on the support of X is
F(x) = P(X ≤ x) =
Γ(β, x/α)
Γ(β)
where
Γ(s, x) =
Z x
x > 0,
t s−1 e−t dt
0
for s > 0 and x > 0 is the incomplete gamma function and
Γ(s) =
Z ∞
t s−1 e−t dt
0
for s > 0 is the gamma function. The survivor function on the support of X is
S(x) = P(X ≥ x) = 1 −
Γ(β, x/α)
Γ(β)
x > 0.
The hazard function on the support of X is
f (x)
xβ−1 e−x/α
h(x) =
=
S(x) (Γ (β) − Γ (β, x/α)) αβ Γ (β)
1
x > 0.
The cumulative hazard function on the support of X is
Γ(β, x/α)
H(x) = − ln S(x) = − ln 1 −
Γ(β)
x > 0.
There is no closed-form expression for the inverse distribution function. The moment generating
function of X is
1
M(t) = E etX = (1 − αt)−β
t< .
α
The characteristic function of X is
φ(t) = E eitX = (1 − αit)−β
The population mean, variance, skewness, and kurtosis of X are
"
3 #
2
X −µ
E[X] = αβ
V [X] = α2 β
E
=p
σ
β
t<
1
.
α
E
"
X −µ
σ
4 #
6
= 3+ .
β
For X1 , X2 , . . . , Xn mutually independent gamma(α, β) random variables, the method of moments
for α and β are
α̂ = s2 /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.
2