Can anyone prove this statement?

  • 1
    $\begingroup$ Hi Havertz, welcome to CrossValidated. You will need to be a bit more specific in your question to make it a good one. Correlation means dependence, so unless you are looking for a proof of a more specific statement, this is simply a definition as it's phrased. $\endgroup$
    – Fato39
    Sep 13 '21 at 7:19
  • $\begingroup$ Hi Fato, Thanks for your reply! I am actually wondering if x and y, two random variables must be dependent to each other if they are non-zero correlated? $\endgroup$
    – Havertz
    Sep 13 '21 at 7:22
  • $\begingroup$ Yes, a non-zero correlation implies some sort of dependence, by definition. As per Wikipedia (see the OR between correlation and dependence): "In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. In the broadest sense correlation is any statistical association, though it commonly refers to the degree to which a pair of variables are linearly related." $\endgroup$
    – Fato39
    Sep 13 '21 at 7:26
  • $\begingroup$ Thanks mate! Do you know if I can actually conduct simulations to prove this statement? $\endgroup$
    – Havertz
    Sep 13 '21 at 7:54
  • 2
    $\begingroup$ @Fato Correlation might mean independence colloquially, but certainly not according to the statistical definitions. The correlation is a combination of central second moments whereas independence is defined purely in terms of probabilities. Thus, there really is something that needs to be proven here, even though many might take it for granted. $\endgroup$
    – whuber
    Sep 13 '21 at 21:20

Correlation and independence, in their technical senses in statistics, are conceptually different and only indirectly related. Correlation is a combination of second central moments of a bivariate random variable while the independence concerns the joint probability distribution. Although the relationship, as expressed in the question, is so familiar and intuitive that we take it for granted, it does merit a rigorous demonstration.

There are some subtleties. Wikipedia gets this wrong where it states, without qualification, that independent variables are uncorrelated. This is not always true: the implication fails when either (or both) of the variables have infinite or zero variance, for then their correlation is undefined, not zero.

I have found a thread here on CV where most of the answers make the same error. A remarkably careful answer in a related thread uses a sophisticated characterization of independence to make and prove a correct formulation of the relationship. Here I resort to a different approach based on conditional expectations.

Suppose $(X,Y)$ are independent. One definition is that the distribution of $X$ is the same no matter what value $Y$ might have. This implies that the conditional expectation, if it exists, is constant, whence

$$E[X] = E[X\mid Y].$$

Using the most basic properties of conditional expectation we may deduce

$$E[XY] = E[E[XY]\mid Y] = E[E[X\mid Y]Y] = E[E[X]Y] = E[X]E[Y].$$

When the correlation $\rho(X,Y)$ exists, both variances are finite and nonzero; and all second moments of $(X,Y)$ are finite, whence $E[X]$ and $E[Y]$ exist and are finite. A formula for the correlation is

$$\rho(X,Y) = \frac{E[XY]-E[X]E[Y]}{\sqrt{\operatorname{Var}(X)\operatorname{Var}{Y}}} = \frac{0}{\sqrt{\operatorname{Var}(X)\operatorname{Var}{Y}}} = 0.$$

This shows that

Independent variables either have no defined correlation or else their correlation is zero.

When the correlation is nonzero, this logically implies the variables are not independent, QED.


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