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.

Moment-generating function of a gamma distribution ( $M_{X} (t) = (λ / (λ - t))^{r}, t < λ$ )

For the rate-parameterized Gamma distribution with shape

r > 0

and rate

λ > 0

, integration of its probability density gives

E [e^{tX}] = \frac{λ ^{r}}{Γ ( r )} \int_{0}^{\infty} x^{r - 1} e^{- (λ - t) x} d x = (\frac{λ}{λ - t})^{r}

when

t < λ

, by the gamma function integral and a change of variables. The integral is infinite for

t \geq λ

. In particular the moment-generating function is finite on an open interval containing zero, so the uniqueness theorem for moment-generating functions applies. Multiplication of these moment-generating functions proves additivity of independent gamma distributions with a common rate.

Gamma density ( $f (y) = θ^{k} y^{k - 1} e^{- θ y} /Γ (k), y > 0$ )

The probability density function of a Gamma distribution with shape

k > 0

and rate

θ > 0

. Integration against

y^{- j}

gives

E [Y^{- j}] = θ^{j} Γ (k - j) /Γ (k)

when

j < k

; the integral diverges for

j \geq k

. Rate and scale are reciprocal conventions and must be distinguished.

Wilson-Hilferty transformation

Shifted gamma distribution ( $V = k + Y, Y \sim Gamma (α, ν)$ )

A gamma distribution translated by a real constant

k

. With shape

α > 0

and rate

ν > 0

, its support is

(k, \infty)

and its first three cumulants are

k + α / ν

,

α / ν^{2}

, and

2 α / ν^{3}

. Its skewness is

2/ α

. A negative shift permits negative values; introducing a shift does not automatically respect the nonnegative support of an insurance loss.

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.

Two independent exponential variables conditioned on their sum ( $Y ∣ (X + Y = u) \sim Unif (0, u)$ )

If

X, Y

are independent exponential random variables with one common rate

λ > 0

, put

U = X + Y

. The change of variables for probability densities

(x, y) = (u - y, y)

gives

f_{U, Y} (u, y) = λ^{2} e^{- λ u}

on

0 < y < u

. Its marginal density is

λ^{2} u e^{- λ u}

, so the conditional probability density of

Y

given

U = u > 0

is

1/ u

on

(0, u)

. This is the shape-

(1, 1)

case of beta-gamma independence. In particular

Y / U

is uniform on

(0, 1)

and independent of

U

. Equality of the rates is essential: different rates leave an exponential tilt across the conditional interval.

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

.

Gamma prior for an inverse-gamma scale ( $Θ ∣ x \sim Gamma (α + nk, λ + \sum_{i} x_{i}^{- 1})$ )

For inverse-gamma distribution observations of known shape

k

and scale

Θ

, their likelihood function in

Θ

is proportional to

Θ^{nk} exp (- Θ \sum_{i} x_{i}^{- 1})

α

, rate-

λ

gamma distribution conjugate prior therefore gives the displayed Bayesian posterior. The Bayes estimator under squared error loss of the conditional claim expected value

Θ/ (k - 1)

is

(α + nk) / [(k - 1) (λ + \sum_{i} x_{i}^{- 1})]

. It depends on reciprocals, rather than the sample mean of the claims.

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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