# Intuition behind the noncentral chi square as Poisson mixing

It is known (cf. Wikipedia, Noncentral_chi_distribution) that the non-central chi-square distribution with k degrees of freedom is a Poisson weighted mixture of central chi-squared distributions).

There are many proofs but all are basically the same (see same ref section "Derivation of the pdf" for a sketch and here libre stats for a freely accessible complete proof).

I'm looking for the intuition behind this result, that is, a way to interpret and explain it. Otherwise such the result seems kind of accident. Any idea ?

best, G.

The [non-central $$\chi^2$$ distribution] has been [sic] obtained by [Fisher, R. A. (1928). Proc. Roy. Soc. A, 121, 654] as a particular case of the distribution of the multiple correlation coefficient. A purely analytical proof was given by [Tang, P. C. (1938). Statist. Res. Mem. 2, 126]. [...] We give a direct geometrical derivation of the [non-central $$\chi^2$$ distribution].
The geometric derivation by Patnaik uses a series expansion of the density, which is equivalent to the series expansion of the characteristic or moment generating function in the usual quick derivation of the non-central $$\chi^2$$ distribution as a Poisson mixture of central $$\chi^2$$ distributions.