Chi-squared distribution

A chi-squared distributed random variable $𝑋 \sim 𝜒 2 𝑛$ is the sum of squares of $𝑛$ independent and identically distributed ${𝑍 𝑖} 𝑛 𝑖 = 1$ with standard normal distributions. #m/def/prob

𝑋 = 𝑛 \sum 𝑖 = 1 (𝑍 𝑖) 2 𝑍 𝑖 i i d \sim N (0, 1)

This turns out to be a special case of the Gamma distribution, namely $𝑋 \sim G a m m a (𝑛 2, 12)$ .

Proof

Let $𝑋 = 𝑍 2$ for $𝑍 \sim N (0, 1)$ , i.e. $𝑋 \sim 𝜒 21$ . Then

𝐹 𝑋 (𝑥) = ℙ (𝑍 2 \leq 𝑥) = ℙ (- \sqrt 𝑥 < 𝑍 < \sqrt 𝑥) = Φ (\sqrt 𝑥) - Φ (- \sqrt 𝑥) = 2 Φ (\sqrt 𝑥) - 1

thus

𝑓 𝑋 (𝑥) = 𝜑 (\sqrt 𝑥) 𝑥 - 1 / 2 = (1 / 2) 1 / 2 Γ (1 / 2) 𝑥 1 / 2 - 1 e - 𝑥 / 2

so $𝑋 \sim G a m m a (12, 12)$ . Thus by ^Q1, the claim is proven.

Properties

Additional properties

Let ${𝑋 𝑗} 𝑛 𝑗 = 1$ be a random sample of variable independently distributed according to the normal distribution $N (𝜇, 𝜎 2)$ . Then the sample variance is distributed such that

(𝑛 - 1) 𝑆 2 𝑛 𝜎 2 \sim 𝜒 2 𝑛 - 1

#state/tidy | #lang/en | #SemBr