本文所探討的是關於 gamma 及 beta 分佈的刻劃。若X，Y為二非退化的隨機變數，滿足X/Y}|Y~B(p,q) 及E((Y-X)^k|X)=b，則X，Y$均為 gamma分佈，且有相同的尺度參數。另外，我們也將利用一些特別形式的二次條件動差來刻劃 gamma 及 beta分佈。

In this work first we characterize two nondegenerate positive random variables X and Y to be gammally distributed with the same scale parameter under the assumptions X/Y}|Y~B(p,q) and E((Y-X)^k|X)=b for some fixed integer k>1
and constant b. Among others characterizations of the gamma and beta distributions based on some special quadratic forms of conditional moments are also investigated.

1.Introduction...................1
2.Preliminaries..................2
3.Identification.................3
4.Discussion....................13

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