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.

中文翻译：

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非中心卡方的（非中心）卡方混合是（非中心）卡方和相关结果、推论和应用

我们主要的、新颖的结果是非中心卡方分布的某个非中心卡方混合本身就是一个缩放的非中心卡方分布。从这个链接到缩放中心卡方分布的混合表示的已知结果，随后会产生许多进一步的混合结果，包括旧的和新的。这些包括中心F、非中心F和 Libby-Novick 分布的混合结果。主要结果涉及所有具有相同自由度的分布；它还扩展到混合非中心卡方分布的自由度比条件非中心卡方分布的自由度大偶数的情况，并追求进一步的结果。