臺灣博碩士論文加值系統

給定$X$，$Y$為二獨立且非退化的隨機變數，Lukacs (1995) 證明 $X/(X+Y)$ 和 $X+Y$ 獨立，若且唯若 $X$，$Y$ 均有 gamma 分佈，且有相同的尺度參數。
本文中，在 $X/U$ 和 $U$ 獨立且 $X/U$ 為 ${\mathcal Be}(p,q)$ 分佈的假設之下，我們利用 $E(h(U,X)|X)\\=b$ 的條件來刻劃 $(U,X)$ 的分佈。其中，$h$ 被允許為 $U-X$ 的指數函數或是 $U-X$ 的三角函數。我們的結果之一是: 假如 $q=1$，且對於某一正整數 $n$，$E(\sum_{i=1}^n e^{i(U-X)}|X)=b$，其中 $b$ 為一常數，則 $(U,X)$ 的分佈可以決定。一些相關的其他結果也將被提出。

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.

中文摘要 .......................... ii
英文摘要 ......................... iii
1. Introduction .................... 1
2. Preliminaries ................... 2
3. Main results .................... 3
4. Conclusion ..................... 16
References ........................ 17
小傳 .............................. 18

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