It is known (cf. [Wikipedia, Noncentral\_chi\_distribution](https://en.wikipedia.org/wiki/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](https://stats.libretexts.org/Bookshelves/Probability_Theory/Book%3A_Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/05%3A_Special_Distributions/5.09%3A_Chi-Square_and_Related_Distribution) 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.



Intuition behind the noncentral chi square as Poisson mixing
[Patnaik](https://academic.oup.com/biomet/article-abstract/36/1-2/202/200823) writes on p. 203:



> 
> 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.


So, I think there has never been any "intuition behind this result" -- just a lucky and rather simple analytical observation.


