The Gamma distribution is a two-parameter family of continuous probability distributions that generalises the Exponential Distribution. It arises naturally as the sum of independent exponential random variables and is used to model waiting times, lifetimes, and rates.
A random variable \(X\) follows a Gamma distribution with shape parameter \(\alpha > 0\) and rate parameter \(\beta > 0\) if
\[ X \sim \text{Gamma}(\alpha, \beta) \quad \text{or} \quad X \sim \Gamma(\alpha, \beta). \]
Parameterisations: The shape-rate form uses \(\alpha\) (shape) and \(\beta\) (rate); the shape-scale form uses \(\alpha\) and \(\theta = 1/\beta\) (scale). We use shape-rate here.
The probability density function (shape-rate form) is
\[ f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x}, \quad x > 0, \]
where \(\Gamma(\alpha)\) is the gamma function.
Interpretation: The shape \(\alpha\) controls how peaked the density is and the rate \(\beta\) sets the scale; for \(\alpha \leq 1\) the density is strictly decreasing, and for \(\alpha > 1\) it is unimodal.
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from scipy.stats import gamma
sns.set_style('whitegrid')
sns.set_palette('Set2')
x = np.linspace(0, 20, 400)
fig, ax = plt.subplots(figsize=(7, 4))
for a in (1, 2, 5, 9):
ax.plot(x, gamma.pdf(x, a, scale=1),
lw=2, label=rf'$\alpha = {a}$')
ax.set_xlabel('$x$')
ax.set_ylabel('$f(x)$')
ax.set_title(r'Gamma density ($\beta = 1$)')
ax.legend()
plt.show()
The expectation and variance of \(X \sim \text{Gamma}(\alpha, \beta)\) are
\[ \mathbb{E}[X] = \frac{\alpha}{\beta} \quad \& \quad \operatorname{Var}(X) = \frac{\alpha}{\beta^2}. \]
Proof. Using \(\int_0^\infty x^{\alpha} e^{-\beta x} \, dx = \Gamma(\alpha + 1)/\beta^{\alpha + 1}\),
\[ \mathbb{E}[X] = \frac{\beta^\alpha}{\Gamma(\alpha)} \cdot \frac{\Gamma(\alpha + 1)}{\beta^{\alpha + 1}} = \frac{\alpha}{\beta}. \]
Similarly \(\mathbb{E}[X^2] = \alpha(\alpha + 1)/\beta^2\), so \(\operatorname{Var}(X) = \alpha/\beta^2\).
The moment generating function (MGF) of \(X \sim \text{Gamma}(\alpha, \beta)\) is
\[ M_X(t) = \left(\frac{\beta}{\beta - t}\right)^\alpha, \quad t < \beta. \]
The characteristic function of \(X \sim \text{Gamma}(\alpha, \beta)\) is
\[ \varphi_X(t) = \left(\frac{\beta}{\beta - it}\right)^\alpha. \]
If \(X_i \sim \text{Gamma}(\alpha_i, \beta)\) are independent (same rate \(\beta\)), then
\[ \sum_{i=1}^n X_i \sim \text{Gamma}\!\left(\sum_{i=1}^n \alpha_i, \beta\right). \]
Proof. The MGF product gives \(\prod_{i=1}^n (\beta/(\beta - t))^{\alpha_i} = (\beta/(\beta - t))^{\sum_i \alpha_i}\).
In particular, if \(X_1, \ldots, X_n\) are i.i.d. \(\text{Exponential}(\beta)\), then \(X_1 + \cdots + X_n \sim \text{Gamma}(n, \beta)\), so the Gamma distribution models the total of \(n\) independent waiting times.
The total time to wait for \(n\) events in a Poisson process — the sum of \(n\) exponential inter-arrival times — is \(\text{Gamma}(n, \lambda)\).
The lifetime of a system with parallel redundant components can be modelled with a Gamma distribution.
The Gamma distribution is the conjugate prior for the rate parameter of a Poisson or exponential likelihood.