A (non‐central) chi‐squared mixture of non‐central chi‐squareds is (non‐central) chi‐squared and related results, corollaries and applications

Our main, novel, result is that a certain non-central chi-squared mixture of distributions itself scaled distribution. From this and link to known on representation for central distribution, numerous further results, both old new, ensue. These include results F, F Libby–Novick distributions. The main involves all with the same degrees freedom; it also extended case where mixing distribution has freedom an even number larger than conditional consequences pursued.