Chi-Squared Distribution

For a sequence of independent Random Variables that are normally distributed. Then the Random Variable ${\sum }_{j=1}^r\left(\frac{X_j}{{\sigma }_j}\right)$ has distribution ${\chi }_{r,\lambda }^2$ with $r$ Degrees of Freedom and Non-Centrality Parameter $\lambda ={\sum }_{j=1}^r{\left(\frac{{\mu }_j}{{\sigma }_j}\right)}^2.$

When all Expectations in the sequence are the same and $0$, then the distribution has PDF $f(x)=\frac{x^{r/2-1}}{\Gamma (r/2)2^{r/2}}{e}^{-x/2}{1}_{(0,\infty )}(x)$ which is also the density of the Gamma Distribution $\Gamma (r/2,1/2)$.