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.