The chi-squared distribution is a continuous probability distribution that arises as the sum of squares of independent standard normal random variables. It is fundamental to hypothesis testing, confidence intervals, and goodness-of-fit tests.
A random variable \(X\) follows a chi-squared distribution with \(k\) degrees of freedom if
\[ X \sim \chi^2_k \quad \text{or} \quad X \sim \chi^2(k). \]
It is a special case of the Gamma Distribution:
\[ \chi^2_k = \text{Gamma}\!\left(\frac{k}{2}, \frac{1}{2}\right). \]
Using the gamma representation, the probability density function is
\[ f(x) = \frac{1}{2^{k/2}\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}, \quad x > 0. \]
Interpretation: For \(k = 1, 2\) the density is decreasing, while for \(k \geq 3\) it is unimodal and increasingly symmetric as \(k\) grows.
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from scipy.stats import chi2
sns.set_style('whitegrid')
sns.set_palette('Set2')
x = np.linspace(0, 20, 400)
fig, ax = plt.subplots(figsize=(7, 4))
for k in (1, 2, 4, 8):
ax.plot(x, chi2.pdf(x, k), lw=2, label=rf'$k = {k}$')
ax.set_xlabel('$x$')
ax.set_ylabel('$f(x)$')
ax.set_title('Chi-squared density')
ax.set_ylim(0, 0.5)
ax.legend()
plt.show()
The expectation and variance of \(X \sim \chi^2_k\) are
\[ \mathbb{E}[X] = k \quad \& \quad \operatorname{Var}(X) = 2k. \]
Proof. Using the gamma representation with shape \(k/2\) and rate \(1/2\),
\[ \mathbb{E}[X] = \frac{k/2}{1/2} = k, \qquad \operatorname{Var}(X) = \frac{k/2}{(1/2)^2} = 2k. \]
The moment generating function (MGF) of \(X \sim \chi^2_k\) is
\[ M_X(t) = (1 - 2t)^{-k/2}, \quad t < \frac{1}{2}. \]
The characteristic function of \(X \sim \chi^2_k\) is
\[ \varphi_X(t) = (1 - 2it)^{-k/2}. \]
If \(Z_1, \ldots, Z_k\) are i.i.d. \(\mathcal{N}(0, 1)\), then
\[ X = \sum_{i=1}^k Z_i^2 \sim \chi^2_k. \]
Proof. Each \(Z_i^2 \sim \chi^2_1 = \text{Gamma}(1/2, 1/2)\). By additivity of the Gamma Distribution with common rate, the sum is \(\text{Gamma}(k/2, 1/2) = \chi^2_k\).
If \(X_i \sim \chi^2_{k_i}\) are independent, then
\[ \sum_{i=1}^n X_i \sim \chi^2_{k_1 + \cdots + k_n}. \]
The chi-squared statistic compares observed and expected frequencies to test whether data follow a hypothesised distribution.
For a normal sample, \(\frac{(n - 1)S^2}{\sigma^2} \sim \chi^2_{n-1}\), which yields confidence intervals and tests for the population variance.
In a contingency table, the chi-squared test assesses whether two categorical variables are independent.