關於二維之變數，Arnold and Press (1989) 首先提出檢驗兩個條件分配是否滿足相容性的理論。本研究嘗試對n維之變數，探討n個條件分配滿足相容性的檢驗方式；並提出在三維聯合分配下，給定三個條件分配為常態(normal) 時，檢驗此三個條件分配滿足相容性的充分必要條件；最後，並推導出此三個條件分配滿足相容性時，其所對應的聯合機率密度函數之公式。若此三個條件分配其所對應的聯合機率密度函數進一步假設為常態時，檢驗其相容性的充分必要條件可更加以簡化。

Arnold and Press (1989) first provide the theory about the compatibility of two conditional distributions in two dimensions. In this research, we extend the two dimensional cases to the high dimensional cases. In particular, we find the necessary and sufficient conditions of the compatibility of three conditional normal distributions in three dimensions. Furthermore, we also provide a formula to find the joint probability density function when three dimensional conditional normal distributions are compatible. Finally, simple sufficient and necessary conditions are also given when the joint distribution is further assumed to be normal.

[1] Arnold, B. C. and Press, S. J.(1989) Compatible conditional distributions. J.Amer. Statist. Assoc., 84, 152-156.
[2] Yates, R. D. and Goodman, D. J.(2005) Probability and Stochastic Processes (2nd ed), John Wiley ＆ Sons.
[3] 蕭惠玲撰(2010)，二維聯合分配下條件常態分配相容性之探討，國立政治大學應用數學系數學教學碩士在職專班碩士論文。