我们知道零相关并不意味着独立。我对非零相关性是否隐含依赖关系很感兴趣-即,如果对于一些随机变量X和Y的,我们能否总体上说f X ,Y(x ,y )≠ f X(x )f Y(y )?
我们知道零相关并不意味着独立。我对非零相关性是否隐含依赖关系很感兴趣-即,如果对于一些随机变量X和Y的,我们能否总体上说f X ,Y(x ,y )≠ f X(x )f Y(y )?
Answers:
是的,因为
这将是不可能的,如果。所以
问题:没有密度的随机变量会发生什么?
\implies
产生 看起来比\rightarow
产生更好。
令和Y表示随机变量,使得E [ X 2 ]和E [ Y 2 ] 是有限的。然后,E [ X Y ],E [ X ]和E [ Y ]都是有限的。
Restricting our attention to such random variables, let denote the statement that and are independent random variables and the statement that and are uncorrelated random variables, that is, . Then we know that implies , that is, independent random variables are uncorrelated random variables. Indeed, one definition of independent random variables is that equals for all measurable functions and ). This is usually expressed as
correlated random variables are dependent random variables.
If , or are not finite or do not exist, then it is not possible to say whether and are uncorrelated or not in the classical meaning of uncorrelated random variables being those for which . For example, and could be independent Cauchy random variables (for which the mean does not exist). Are they uncorrelated random variables in the classical sense?
Here a purely logical proof. If then necessarily , as the two are equivalent. Thus if then . Now replace with independence and with correlation.
Think about a statement "if volcano erupts there are going to be damages". Now think about a case where there are no damages. Clearly a volcano didn't erupt or we would have a condtradicition.
Similarly, think about a case "If independent , then non-correlated ". Now, consider the case where are correlated. Clearly they can't be independent, for if they were, they would also be correlated. Thus conclude dependence.