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

Definition: Let $Z_1, \ldots, Z_k$ be independent random variables where each of them is following a standard normal distribution:

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

Define the random variable $X$ as the sum of all squared $Z_i$:

\[\label{eq:X} X = \sum_{i=1}^{k} Z_i^2 \; .\]

Then, the variable $X$ is said to follow a chi-squared distribution with $k$ degrees of freedom:

\[\label{eq:wish} X \sim \chi^{2}(k)\]

where $k > 0$.

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