关于随机变量独立性的研究

Research on the Independence of Random Variables

研究了随机变量独立性问题，给出了一个判断随机变量独立性的充分必要条件的定１理．这个定理为：设（Ｘ，Ｙ）是连线型随机变量，它们的联合密度函数为ｆ（Ｘ，ｙ），其中：ａ≤ｘ≤ｂｃ≤ｙ≤ｄ则随机变量Ｘ，Ｙ相互独立的充分必要条件为：１°存在连续函数ｈ（ｘ）ｇ（ｙ），使ｆ（ｘ，ｙ）＝ｈ（ｘ）ｇ（ｙ）几乎处处成立．２°ａｂ，ｃ，ｄ是与ｘ，ｙ无关的常量．这个定理对研究随机变量的独立性是很方便的．

Under the condition of sufficient and mecessary reason a theorem for disccusing independece of continuous random variables is given as follows:Let (X, Y) be continuous random variables. Its joint density function is f(x, y), here a≤x≤b, and c≤y≤d. Then sufficient and necessary condition for continuous random vairables X, Y independent of each other will be:1. At presence of continuous functions h(x) and g(y),f(x, y)=h(x)g(y) exist almost everywhere.2. a, b, c, d are constants having nothing tO do with X, y.