Chi-square distribution
Story
The sum of the squares of \(\nu\) Standard-Normally distributed variables is Chi-square distributed. The distribution generalizes to non-integer \(\nu\).
Example
The Chi-square distribution is seldom useful in modeling contexts; it is typically used in null hypothesis significance testing where it arises in analytical treatments of specific tests.
Parameter
The Chi-square distribution has a single positive parameter, \(\nu\).
Support
The Chi-square distribution is supported on the set of positive real numbers.
Probability density function
\[\begin{align}
f(y;\nu) = \frac{1}{\Gamma(\nu/2)}\,\frac{(y/2)^{\nu/2}}{y} \,\mathrm{e}^{-y/2},
\end{align}\]
where \(\Gamma(x)\) is the gamma function.
Cumulative distribution function
\[\begin{align}
F(y;\nu) = P(\nu/2, y/2),
\end{align}\]
Moments
Mean: \(\nu\)
Variance: \(2\nu\)
Usage
Package |
Syntax |
|---|---|
NumPy |
|
SciPy |
|
Distributions.jl |
|
Stan |
|