关于相关系数的深层讨论

Exploration of Correlation Coefficient

相关系数XρY是描述二维随机变量(X,Y)之间线性相关关系的一个重要数字特征。当XρY=0时,表明X和Y之间不存在线性关系,可能存在非线性函数关系等。作者在《概率论与数理统计》教学实践中发现不少学生想当然地认为:当X和Y真正存在非线性函数关系时,相关系数XρY=0。文章以正态分布、均匀分布为例论证了相关系数XρY并不总是为零,甚至XρY≈1的事实,并指出相关系数的大小与X的分布有关。

Linear correlation coefficient Pxr is one of important numerical characteristics to describe the correlation between two-dimensional random variables （X, Y）. PXY indicates no linear relation exists between X and Y and some nonlinear function relation may exists. In teaching ＂ Probability Theory and Mathematical Statistics＂, the author finds many students take it for granted that PXY is equal to zero when Y is a nonlinear function of random variable X. In fact,it is shown that PXY is not always equal to zero, even close to ＋ 1, which is connected with the distribution of X. In this article, the linear correlation coefficinet PXY is discussed in detail when Y = X： and X is normal distribution or uniform distribution.