在第一章，先回顧目前一些二元分佈和二元機率結合函數(Bivariate Copulas)相關的定義、主要定理，及目前的發展概況。接著介紹一些常見的機率結合函數，如AC族、FGM族及常態機率結合函數族。由回顧知道，經由二元機率合函數以建立二元分佈是一項很有用的工具，尤其是邊際分佈為特定分佈時，如邊際分佈為同類型時之二元指數、二元迦瑪等等。
在第二章，本文提出一個新方法以加大二元RULF機率結合函數族，並由調整加大後函數族的參數來提高|ρs|，並加大模型在實際資料的適用範圍。此外地，舉出一些由二元機率結合函所數造出之新的二元指數分配來當例子。
在第三章，經觀察Farlie(1960)及RULF族，領悟出一些新的二元機率結合函數族，但這些函數不屬於AC族、RULF族及目前一些已知的機率結合函數族。這些新的機率結合函數族，參數在某些特定值時，有正象相依(Positive Quadrant Dependence，PQD)的性質。

In chapter one, we review recent developments on bivariate distributions and bivariate copulas. We also introduce some bivariate copulas, such as Archimedean copulas, Farlie-Gumbel-Morgenstern copulas  and Normal copulas . It is obvious to see that bivariate copulas is a very useful tool for constructing bivariate distributions, especially when the marginal distributions are given of the same type of distribution such as bivariate exponential distribution and bivariate gamma distribution.
In chapter two, we provide some new copulas which may be considered as some extension of RULF and this new copulas increases |ρs| and also increases its applicability for real data. We give some bivariate exponential distributions which are constructed by the proposed bivariate copulas.
Finally, in chapter three, we propose some new bivariate copulas which don’t belong to AC family nor RULF family nor other family so far it is known. Those proposed copulas possess some property of PQD.

第一章  二元分佈介紹	1
1.1  緒論	1
1.2  二元分佈相關簡介與定義	3
1.2A  機率結合函數族群	3
1.2B  FGM(Farlie－Gumbel－Morgenstern)族	8
第二章  一些二元機率結合函數族之推廣	13
2.1  二元RLUF族機率結合函數之推廣	13
第三章  一些新的二元機率結合函數族	22
3.1  由Farlie(1960)所觀得之函數族	25
3.2  由RLUF所觀得之函數族	25
附錄一  RLUF族之參數估計	27

表  圖  目  錄
表2.1 定理2.3中(1)之特殊函數	15
表2.2 定理2.3中(2)之特殊函數	16
表3.1 當θ=1－e及θ=e，(3.3)式之一階動差及Spearman相關係數	23
表3.2 當θ=－1及θ=1，(3.5)式之一階動差及Spearman相關係數	23
圖3.1  θ=e時，(3.3)之機率密度函數圖	24
圖3.2  θ=1時，(3.5)之機率密度函數圖	24
圖3.3  θ=1時，(3.10)之機率密度函數圖

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