Index: The Book of Statistical ProofsProbability Distributions ▷ Univariate continuous distributions ▷ Chi-squared distribution ▷ Definition

Definition: Let $X_{1}, …, X_{k}$ be independent random variables where each of them is following a standard normal distribution:

\[\label{eq:snorm} X_{i} \sim \mathcal{N}(0,1) \quad \text{for} \quad i = 1, \ldots, n \; .\]

Then, the sum of their squares follows a chi-squared distribution with $k$ degrees of freedom:

\[\label{eq:chi2} Y = \sum_{i=1}^{k} X_{i}^{2} \sim \chi^{2}(k) \quad \text{where} \quad k > 0 \; .\]

The probability density function of the chi-squared distribution with $k$ degrees of freedom is

\[\label{eq:chi2-pdf} \chi^{2}(x; k) = \frac{1}{2^{k/2}\Gamma (k/2)} \, x^{k/2-1} \, e^{-x/2}\]

where $k > 0$ and the density is zero if $x \leq 0$.


Metadata: ID: D100 | shortcut: chi2 | author: kjpetrykowski | date: 2020-10-13, 01:20.