Noncentral chi-squared Distribution

The distribution of ${\sum }_{i=1}^{\nu }{\left(Z_i+{\delta }_i\right)}^2$ where $Z_i$ are independent standard normal variables and ${\delta }_i$ are constants. $\lambda ={\sum }_{i=1}^{\nu }{\delta }_i^2>0.$ (In communications it is called the Marcum-Q function). It can be thought of as a Generalized Rayleigh-Rice distribution.

The two shape parameters are $\nu$, a positive integer, and $\lambda$, a positive real number. The support is $x\ge 0$.

where $I_{\nu }(y)$ is a modified Bessel function of the first kind.

References

• "Noncentral chi-squared distribution", Wikipedia https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution

Implementation: scipy.stats.ncx2