Chi-Squared Distribution

For a sequence of independent Random Variables that are normally distributed. Then the Random Variable

∑j=1r(Xjσj)\sum_{j=1}^{r}{\left(\frac{X_j}{\sigma_j}\right)}

has distribution

χr,λ2\chi^2_{r,\lambda}

with rr Degrees of Freedom and Non-Centrality Parameter

λ=∑j=1r(μjσj)2.\lambda=\sum_{j=1}^{r}{\left(\frac{\mu_j}{\sigma_j}\right)^2}.

When all Expectations in the sequence are the same and 00, then the distribution has PDF

f(x)=xr/2−1Γ(r/2)2r/2e−x/21(0,∞)(x)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 Γ(r/2,1/2)\Gamma(r / 2,1 / 2).