Gamma Distribution — Sum of Exponential Variables

Foundations of Statistics

Flexible Modeling of Positive Data

The gamma distribution extends the exponential to model waiting times for multiple events and right-skewed positive data. Its flexibility makes it ideal for insurance claims, rainfall amounts, and survival analysis.

Insurance — Modeling claim sizes and aggregate losses in actuarial science
Meteorology — Predicting rainfall amounts and drought durations
Healthcare — Survival times in clinical trials and time-to-event data

When data is positive and skewed, the gamma distribution provides the natural framework.

Core Concepts

The gamma distribution generalizes the exponential distribution. It models the waiting time until the $\alpha$ -th event in a Poisson process and serves as a flexible model for right-skewed, positive-valued data.

DfGamma Distribution

A continuous random variable $X$ has a gamma distribution with shape parameter $\alpha > 0$ and rate parameter $\beta > 0$ if its pdf is:

f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x}, \quad x > 0

Written $X \sim \text{Gamma}(\alpha, \beta)$ . It represents the sum of $\alpha$ independent $\text{Exp}(\beta)$ variables (when $\alpha$ is integer).

The Gamma Function

DfThe Gamma Function

The Gamma function is defined for $\alpha > 0$ by:

\Gamma(\alpha) = \int_0^{\infty} t^{\alpha - 1} e^{-t} \, dt

Key properties:

$\Gamma(n) = (n-1)!$ for positive integers $n$
$\Gamma(\alpha + 1) = \alpha \, \Gamma(\alpha)$ (functional equation)
$\Gamma(1/2) = \sqrt{\pi}$
$\Gamma(n + 1/2) = \frac{(2n)!}{4^n n!} \sqrt{\pi}$

ThProof that Γ(n) = (n-1)! for Integer n

Base case: $\Gamma(1) = \int_0^\infty e^{-t} dt = 1 = 0!$

Inductive step: Assume $\Gamma(k) = (k-1)!$ . Then:

\Gamma(k+1) = \int_0^\infty t^k e^{-t} dt

Integration by parts with $u = t^k$ , $dv = e^{-t} dt$ :

= \left[-t^k e^{-t}\right]_0^\infty + k \int_0^\infty t^{k-1} e^{-t} dt = 0 + k \cdot \Gamma(k) = k \cdot (k-1)! = k!

PDF of Gamma Distribution

f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}, \quad x > 0

Here,

$\alpha$ =Shape parameter
$\beta$ =Rate parameter
$\Gamma(\alpha)$ =Gamma function (generalization of factorial)

Derivation of Mean and Variance

ThDerivation of E[X] and Var(X)

Mean:

E[X] = \int_0^{\infty} x \cdot \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x} \, dx = \frac{\beta^\alpha}{\Gamma(\alpha)} \int_0^{\infty} x^{\alpha} e^{-\beta x} \, dx

Substitute $u = \beta x$ , $dx = du/\beta$ :

= \frac{\beta^\alpha}{\Gamma(\alpha)} \cdot \frac{\Gamma(\alpha+1)}{\beta^{\alpha+1}} = \frac{\alpha \, \Gamma(\alpha)}{\beta \, \Gamma(\alpha)} = \frac{\alpha}{\beta}

Second moment:

E[X^2] = \frac{\beta^\alpha}{\Gamma(\alpha)} \int_0^{\infty} x^{\alpha+1} e^{-\beta x} \, dx = \frac{\beta^\alpha}{\Gamma(\alpha)} \cdot \frac{\Gamma(\alpha+2)}{\beta^{\alpha+2}} = \frac{(\alpha+1)\alpha}{\beta^2}

Variance:

\text{Var}(X) = \frac{\alpha(\alpha+1)}{\beta^2} - \frac{\alpha^2}{\beta^2} = \frac{\alpha}{\beta^2}

Gamma Mean and Variance

E[X] = \frac{\alpha}{\beta}, \quad \text{Var}(X) = \frac{\alpha}{\beta^2}

Here,

$\alpha$ =Shape parameter
$\beta$ =Rate parameter

MGF and Additivity

MGF of Gamma Distribution

M_X(t) = \left(\frac{\beta}{\beta - t}\right)^\alpha, \quad t < \beta

Here,

$t$ =Transform variable
$\alpha$ =Shape parameter
$\beta$ =Rate parameter

ThSum of Independent Gammas with Same Rate

If $X_1 \sim \text{Gamma}(\alpha_1, \beta)$ and $X_2 \sim \text{Gamma}(\alpha_2, \beta)$ are independent, then:

X_1 + X_2 \sim \text{Gamma}(\alpha_1 + \alpha_2, \beta)

Proof: $M_{X_1+X_2}(t) = M_{X_1}(t) \cdot M_{X_2}(t) = \left(\frac{\beta}{\beta-t}\right)^{\alpha_1} \cdot \left(\frac{\beta}{\beta-t}\right)^{\alpha_2} = \left(\frac{\beta}{\beta-t}\right)^{\alpha_1+\alpha_2}$

This is the MGF of $\text{Gamma}(\alpha_1+\alpha_2, \beta)$ .

Important: This additivity property requires the same rate $\beta$ . For different rates, the sum does not have a gamma distribution.

Special Cases

Special Cases of the Gamma

$\alpha = 1$ : $\text{Gamma}(1, \beta) = \text{Exp}(\beta)$
$\alpha = n/2, \beta = 1/2$ : $\text{Gamma}(n/2, 1/2) = \chi^2_n$ (chi-squared with $n$ degrees of freedom)
$\alpha = k, \beta = 1$ : $\text{Gamma}(k, 1) = \text{Erlang}(k)$ — sum of $k$ i.i.d. $\text{Exp}(1)$ variables
As $\alpha \to \infty$ : $\text{Gamma}(\alpha, \beta) \to N(\alpha/\beta, \alpha/\beta^2)$ by the CLT (sum of $\alpha$ i.i.d. exponentials)

Chi-Squared Connection

ThChi-Squared as a Special Case of Gamma

The chi-squared distribution with $\nu$ degrees of freedom is:

\chi^2_\nu = \text{Gamma}\left(\frac{\nu}{2}, \frac{1}{2}\right)

Proof: If $Z_1, \ldots, Z_\nu$ are i.i.d. $N(0,1)$ , then $Q = Z_1^2 + \cdots + Z_\nu^2 \sim \chi^2_\nu$ .

Each $Z_i^2$ has pdf: $f_{Z^2}(x) = \frac{1}{\sqrt{2\pi x}} e^{-x/2} = \frac{(1/2)^{1/2}}{\Gamma(1/2)} x^{1/2 - 1} e^{-x/2}$

This is $\text{Gamma}(1/2, 1/2)$ . By additivity: sum of $\nu$ such variables is $\text{Gamma}(\nu/2, 1/2) = \chi^2_\nu$ .

Log-Gamma Distribution

The Log-Gamma

If $X \sim \text{Gamma}(\alpha, \beta)$ , then $Y = \ln X$ follows a log-gamma distribution. As $\alpha \to \infty$ , $Y - E[Y]$ approaches a normal distribution. For $\alpha = 1$ (exponential), $\ln X$ follows the Gumbel distribution.

Worked Example

Example: Claim Processing Time

An insurance company finds that claim processing times follow $\text{Gamma}(\alpha = 3, \beta = 0.5)$ (in hours).

Mean processing time: $E[X] = \alpha/\beta = 3/0.5 = 6$ hours

Variance: $\text{Var}(X) = \alpha/\beta^2 = 3/0.25 = 12$ hours $^2$

SD: $\sigma = \sqrt{12} \approx 3.46$ hours

Probability that a claim takes more than 10 hours:

$P(X > 10) = 1 - F(10)$ where $F$ is the gamma CDF. Using the CDF integral or tables:

P(X > 10) = P(\text{Poisson}(0.5 \times 10) \leq 2) = e^{-5}(1 + 5 + 12.5) = 18.5 e^{-5} \approx 0.125

(The relationship $P(X > x) = P(N(x) < \alpha)$ where $N \sim \text{Poisson}(\beta x)$ holds for integer $\alpha$ .)

Specific Applications

Insurance and finance — Aggregate claim amounts, ruin probability, and loss modeling.
Hydrology — Rainfall amounts, flood frequency analysis (often as gamma or Pearson Type III).
Bayesian statistics — Conjugate prior for the Poisson rate parameter.
Queueing theory — Erlang distributions model total service time for multiple sequential tasks.

Key Takeaways

Summary: Gamma Distribution

Generalizes the exponential; models waiting time for $\alpha$ -th event
PDF: $f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}$
Mean: $\alpha/\beta$ , Variance: $\alpha/\beta^2$
MGF: $M_X(t) = (\beta/(\beta-t))^\alpha$ ; sum of same-rate gammas is gamma
$\alpha = 1$ gives Exponential; $\alpha = n/2, \beta = 1/2$ gives Chi-squared
CLT: approaches $N(\alpha/\beta, \alpha/\beta^2)$ as $\alpha \to \infty$
The Gamma function $\Gamma(\alpha)$ generalizes factorial: $\Gamma(n) = (n-1)!$

Gamma Distribution — Sum of Exponential Variables

Gamma Distribution — Sum of Exponential Variables

Flexible Modeling of Positive Data

Core Concepts

DfGamma Distribution

The Gamma Function

DfThe Gamma Function

ThProof that Γ(n) = (n-1)! for Integer n

PDF of Gamma Distribution

Derivation of Mean and Variance

ThDerivation of E[X] and Var(X)

Gamma Mean and Variance

MGF and Additivity

MGF of Gamma Distribution

ThSum of Independent Gammas with Same Rate

Special Cases

Chi-Squared Connection

ThChi-Squared as a Special Case of Gamma

Log-Gamma Distribution

Worked Example

Specific Applications

Key Takeaways

Summary: Gamma Distribution

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