Noncentral Chi-squared Distribution

Definition

Let $X_1,X_2,\dots ,X_k$ be $k$ independent normal distributions $N({\mu }_i,{\sigma }^2)$, then $Y:=\frac{{\sum }_{i=1}^k{X}_i^2}{{\sigma }^2}\sim {\chi }^2(k,\lambda )$ where $k\in \mathbb{N}$ is degree of freedom, and $\lambda :=\sum _{i=1}^k\frac{{\mu }_i^2}{{\sigma }^2}\in {\mathbb{R}}^{+}$ is non-centrality parameter.

Properties

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MGF

$\frac{1}{(1-2t{)}^{k/2}}\exp \left(\frac{\lambda t}{1-2t}\right)$