Editing Chi-squared distribution (section)

=== Sample mean ===
The sample mean of <math>n</math> [[i.i.d.]] chi-squared variables of degree <math>k</math> is distributed according to a gamma distribution with shape <math>\alpha</math> and scale <math>\theta</math> parameters:
:<math> \overline X = \frac{1}{n} \sum_{i=1}^n X_i \sim \operatorname{Gamma}\left(\alpha=n\, k /2, \theta= 2/n \right) \qquad \text{where } X_i \sim \chi^2(k)</math>

[[#Asymptotic properties|Asymptotically]], given that for a shape parameter <math> \alpha </math> going to infinity, a Gamma distribution converges towards a normal distribution with expectation <math> \mu = \alpha\cdot \theta </math> and variance <math> \sigma^2 = \alpha\, \theta^2 </math>, the sample mean converges towards:

<math style="block"> \overline X \xrightarrow{n \to \infty} N(\mu = k, \sigma^2 = 2\, k /n ) </math>

Note that we would have obtained the same result invoking instead the [[central limit theorem]], noting that for each chi-squared variable of degree <math>k</math> the expectation is <math> k </math> , and its variance <math> 2\,k </math> (and hence the variance of the sample mean <math> \overline{X}</math> being <math> \sigma^2 = \frac{2k}{n} </math>).