# Probability density function (pdf) of normal sample variance ($S^2$)

I need to know the formula for the pdf of $S^2$.

I know this:

$$\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1} \>,$$

but I want to state the correct formula for the pdf of $S^2$, not $(n-1)S^2/\sigma^2$.

Any thoughts?

• Is this an homework? Do you know the pdf of a $\chi^2_{n-1}$ distribution? – Xi'an Feb 3 '12 at 17:33
• See 2nd bullet under en.wikipedia.org/wiki/Gamma_distribution#Specializations – onestop Feb 3 '12 at 17:37
• It is for a theory class, but we have no homework. He mentioned it would be a good excerise to prove that (X-bar, S^2) are sufficient statistics for (mu, sigma^2) using the ratio method (not factorization theorem). Yes, I know the pdf of a χ2n−1 distribution. – Patrick Feb 3 '12 at 17:37
• Here is a hint (sorry, had a typo in the previous version): $$\mathbb P(S^2 \leq s) = \mathbb P( (n-1)S^2/\sigma^2 \leq (n-1) s / \sigma^2) = \int_0^{(n-1)s/\sigma^2} \frac{u^{(n-3)/2}e^{-u/2}}{2^{(n-1)/2}\Gamma\big(\frac{n-1}{2}\big)}\,\mathrm{d}u \>.$$ Now, use the fundamental theorem of calculus. – cardinal Feb 3 '12 at 17:43
• What is $S^2$? There are competing definitions for sample variance. – wolfies Mar 18 '16 at 18:25

Given $\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1} \>,$

and the fact that a chi-squared($\nu$) is a Gamma($\frac{\nu}{2},2$), (under the scale parameterization) then

$S^2 = \frac{(n-1)S^2}{\sigma^2}\cdot \frac{\sigma^2}{(n-1)}\sim \text{Gamma}(\frac{(n-1)}{2},\frac{2\sigma^2}{(n-1)})$

If you need a proof, it should suffice to show that the relationship between chi-square and gamma random variables holds and then follow the scaling argument here. This relationship is pretty much verifiable by inspection.

The pdf is as follows:

\begin{equation} f(x) = \frac{\left(\frac{\nu}{2\, \sigma^{2}}\right)^{\frac{\nu}{2}}}{\Gamma\left(\frac{\nu}{2}\right)}\, x^{\frac{\nu}{2}-1}\, \exp\left\{-x\, \frac{\nu}{2\, \sigma^{2}}\right\} \end{equation}

$\nu \equiv \text{degrees of freedom}= N-1$, where $N$ is the sample size.

$\sigma \equiv \text{standard deviation of the parent distribution}$.