Noncentral Chi-squared Distribution - Derivation of The Pdf

Derivation of The Pdf

The derivation of the probability density function is most easily done by performing the following steps:

1. First, assume without loss of generality that . Then the joint distribution of is spherically symmetric, up to a location shift.
2. The spherical symmetry then implies that the distribution of depends on the means only through the squared length, . Without loss of generality, we can therefore take and .
3. Now derive the density of (i.e. k=1 case). Simple transformation of random variables shows that :
   where is the standard normal density.
4. Expand the cosh term in a Taylor series. This gives the Poisson-weighted mixture representation of the density, still for k=1. The indices on the chi-squared random variables in the series above are 1+2i in this case.
5. Finally, for the general case. We've assumed, without loss of generality, that are standard normal, and so has a central chi-squared distribution with (k-1) degrees of freedom, independent of . Using the poisson-weighted mixture representation for, and the fact that the sum of chi-squared random variables is also chi-squared, completes the result. The indices in the series are (1+2i)+(k-1) = k+2i as required.