| Summary: | 碩士 === 國立高雄大學 === 統計學研究所 === 94 === Given two independent non-degenerate positive random variables $X$ and $Y$, Lukacs (1955) proved that $X/(X+Y)$ and $X+Y$ are independent if and only if $X$ and $Y$ are gamma distributed with the same scale parameter.
In this work, under the assumption $X/U$ and $U$ are independent, and $X/U$ has a ${\mathcal Be}(p,q)$ distribution, we characterize the distribution of $(U,X)$ by the condition $E(h(U,X)|X)=b$, where $h$ is allowed to be an exponential function or trigonometric function of $U-X$. Among others, we prove if $q=1$, and for some positive integer $n$, $E(\sum_{i=1}^n e^{i(U-X)}|X)=b$, where $b$ is a constant, then the distribution of $(U,X)$ can be determined. Some other related results are also presented.
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