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