 Chi-squared distribution

A Chi-squared distribution with $\nu >0$ statistical degrees of freedom is a Gamma distribution with shape $\nu /2$ and rate $1/2$. Its Probability density function isFor a positive integer $k$, a ${\chi }_k^2$ variable is the sum of squares of $k$ independent standard normal random variables. The gamma definition extends the law to noninteger positive degrees of freedom; it does not require a fractional number of Gaussian coordinates.

Its mean is $\nu$, its variance is $2\nu$, and its Moment-generating function is $(1-2t{)}^{-\nu /2}$ for $t<1/2$. These follow directly from the Gamma density or by multiplying the squared-normal transforms in the integer case.

The law appears when Gaussian deviations are aggregated as a squared length. For iid observations $X_1,\dots ,X_n\sim N(\mu ,{\sigma }^2)$ with $n\ge 2$ and $\sigma >0$, the Sample variance $S^2=(n-1{)}^{-1}{\sum }_i(X_i-\overline{X}{)}^2$ satisfies $(n-1)S^2/{\sigma }^2\sim {\chi }_{n-1}^2$. Centering removes the one-dimensional sample-mean direction, leaving $n-1$ independent Gaussian coordinates after an orthogonal change of basis. This exact normal-sampling law underlies confidence intervals for a variance; it is not a universal variance-sampling formula for arbitrary data.