Gamma distribution

The gamma distribution with shape

a > 0

and rate

r > 0

f (x) = \frac{r ^{a}}{Γ ( a )} x^{a - 1} e^{- r x}, x > 0.

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Beta-gamma independence ( $X / (X + Y) ⊥ (X + Y)$ )

If

X

and

Y

are independent random variables with Gamma distributions of shapes

a, b > 0

and the same rate

θ > 0

, then

U = X / (X + Y)

has the Beta distribution with parameters

(a, b)

, and

V = X + Y

has the Gamma distribution of shape

a + b

and rate

θ

. Moreover

U

and

V

are independent. The inverse change of variables

(x, y) = (uv, (1 - u) v)

has absolute Jacobian determinant

v

, which makes the joint probability density factorize on the product domain

0 < u < 1

,

v > 0

. The common rate is essential for this factorization.

Additivity of independent gamma distributions with a common rate

If independent random variables

X_{i}

have gamma distributions with shapes

a_{i}

and one common rate

r

, then

\sum_{i = 1}^{n} X_{i} \sim Gamma (\sum_{i = 1}^{n} a_{i}, r) .

Their Laplace transforms of nonnegative random variables multiply to

(r / (r + s))^{\sum_{i} a_{i}}

, which is the transform of the displayed gamma distribution; uniqueness of the Laplace transform proves the claim.

Gamma exponential dispersion family

ϕ = 1/ a

,

θ = - r / a < 0

, and

b (θ) = - lo g (- θ)

puts the gamma density into exponential dispersion family form

f (y; θ, ϕ) = a (y, ϕ) exp (\frac{y θ - b ( θ )}{ϕ}), a (y, ϕ) = \frac{y ^{1/ ϕ - 1}}{Γ ( 1/ ϕ ) ϕ ^{1/ ϕ}} .

Its mean is

a / r

and its variance is

a / r^{2}

.

Cumulant-generating function of a gamma distribution ( $K_{Y} (t) = - a lo g (1 - t / r)$ )

For a gamma random variable of shape

a

and rate

r

,

K_{Y} (t) = lo g E [e^{t Y}] = - a lo g (1 - \frac{t}{r}), t < r .

Inverse-gamma distribution

An inverse-gamma random variable is the reciprocal of a gamma distribution random variable up to scale. In shape-scale parameters its density is proportional to

x^{- a - 1} e^{- b / x}

on

x > 0

.

Erlang distribution

The sum of

n

independent exponential variables of rate

r

has the Erlang distribution with density

f_{n} (x) = \frac{r ^{n} x ^{n - 1}}{( n - 1 )!} e^{- r x}, x > 0.

Convolving with one more exponential proves the formula inductively.

Sampling an integer-shape gamma distribution

If

m

is a positive integer and

U_{1}, \dots, U_{m}

are independent uniform variables, then

- \frac{1}{r} \sum_{j = 1}^{m} lo g U_{j}

has the gamma distribution with shape

m

and rate

r

, because each summand is exponential with rate

r

.

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The Gamma distribution is a continuous probability distribution defined by two parameters: shape (often denoted as \( k \) or \( \alpha \)) and scale (denoted as \( \theta \) or \( \beta \)). It is widely used in various fields, including statistics, finance, and engineering, due to its ability to model waiting times and processes that are characterized by events that occur independently at a constant average rate.

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