Noncentral Chi-Squared Distribution
Definition
The continuous probability distribution that has the probability density function as follows with respect to degrees of freedom $r > 0$ and non-centrality $\lambda \ge 0$ is called the Noncentral Chi-squared Distribution. $$ f(x) = {{ 1 } \over { 2 }} e^{- \left( x + \lambda \right) / 2 } \left( {{ x } \over { \lambda }} \right)^{k/4 - 1/2} I_{r/2 - 1} \left( \sqrt{\lambda x} \right) \qquad, x \in (0,\infty) $$
- $I_{\nu}$ represents the $\nu$th order modified Bessel function of the first kind, and its appearance in directional statistics as explained in why the modified Bessel function of the first kind appears in directional statistics is purely coincidental and has nothing to do with the formula itself1.
Explanation
As the name suggests, the noncentral chi-squared distribution is a generalization of the chi-squared distribution, meaning it applies to independent random variables $$ X_{k} \sim N \left( \mu_{k} , 1^{2} \right) \qquad , k = 1 , \cdots , r $$ that follow a normal distribution when they are defined as $\lambda = \sum_{k=1}^{r} \mu_{k}^{2}$. It implies the following meaning in $Y$. $$ Y = \sum_{k=1}^{r} X_{k}^{2} \sim \chi^{2} \left( r , \lambda \right) $$ That is, $\lambda \ne 0$ does not mean the chi-squared distribution itself, but indicates that the centers of the normal distributions being squared and summed are not $0$.