非零相关是否意味着依赖?


17

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XY的,我们能否总体上说f X Yx y f Xx f Yy Corr(X,Y)0XYfX,Y(x,y)fX(x)fY(y)

Answers:


13

是的,因为

Corr(X,Y)0Cov(X,Y)0

E(XY)E(X)E(Y)0

xyfX,Y(x,y)dxdyxfX(x)dxyfY(y)dy0

xyfX,Y(x,y)dxdyxyfX(x)fY(y)dxdy0

xy[fX,Y(x,y)fX(x)fY(y)]dxdy0

这将是不可能的,如果。所以fX,Y(x,y)fX(x)fY(y)=0,{x,y}

Corr(X,Y)0{x,y}:fX,Y(x,y)fX(x)fY(y)

问题:没有密度的随机变量会发生什么?


1
Alecos,我有一个愚蠢的问题。花式箭头在例如第1行中是什么意思?我想像“暗示”之类的东西,但我不确定。
Sycorax说恢复莫妮卡

2
@ user777您的意思是吗?实际上,它的意思是“暗示”。
Alecos Papadopoulos

仅在非正式参数中使用暗示箭头的原因:暗示箭头是左侧还是右侧关联?
kasterma 2014年

\implies 产生 看起来比\rightarow产生更好。
Dilip Sarwate

14

Y表示随机变量,使得E [ X 2 ]E [ Y 2 ] 是有限的。然后,E [ X Y ]E [ X ]E [ Y ]都是有限的。XYE[X2]E[Y2]E[XY]E[X]E[Y]

Restricting our attention to such random variables, let A denote the statement that X and Y are independent random variables and B the statement that X and Y are uncorrelated random variables, that is, E[XY]=E[X]E[Y]. Then we know that A implies B, that is, independent random variables are uncorrelated random variables. Indeed, one definition of independent random variables is that E[g(X)h(Y)] equals E[g(X)]E[h(Y)] for all measurable functions g() and h()). This is usually expressed as

AB.
But AB is logically equivalent to ¬B¬A, that is,

correlated random variables are dependent random variables.

If E[XY], E[X] or E[Y] are not finite or do not exist, then it is not possible to say whether X and Y are uncorrelated or not in the classical meaning of uncorrelated random variables being those for which E[XY]=E[X]E[Y]. For example, X and Y could be independent Cauchy random variables (for which the mean does not exist). Are they uncorrelated random variables in the classical sense?


3
The nice thing about this answer is that it applies whether or not the random variables in question admit a density function, as opposed to other answers on this thread. This is true due to the fact that expectations can be defined with Stieltjes integrals using the CDF, with no mention of the density.
ahfoss

1

Here a purely logical proof. If AB then necessarily ¬B¬A, as the two are equivalent. Thus if ¬B then ¬A. Now replace A with independence and B 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 X,Y, then non-correlated X,Y". Now, consider the case where X,Y are correlated. Clearly they can't be independent, for if they were, they would also be correlated. Thus conclude dependence.


If you will read my answer carefully, you will see that I too used the argument that you have made in your answer, namely that AB is the same as B¬A.
Dilip Sarwate

@DilipSarwate Edited to reflect that.
Tony
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