Can I determine if two random variables are independent if I know their expected values and the variances?

Question

I have two random variables $X$ and $Y$. I know the expected values $E\left[X\right]$ and $E\left[Y\right]$, as well as their respective variances $V\left[X\right]$ and $V\left[Y\right]$ (I have them actually tabulated from a curve fitting/regression analysis).

If I want to calculate $E\left[X\cdot Y\right]$, I've derived this equation. $$E\left[X\cdot Y\right]=E\left[X\right]\cdot E\left[Y\right]+C\left[X,Y\right]$$

The problem is I don't know the covariance $C\left[X,Y\right]$, but I do know if each random variable is independent, then the covariance is zero. Since I know the expected values and the variances, can demonstrate from the tabulated values the independence of $X$ and $Y$?

Answer 1

Clearly the expectation and variance of a random variable only tells you some things about that random variable, and not about how it may depend on another random variable.

Having two sets of expectations and variances can not indicate anything about how the respective variables depend on each other.

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PS: Even knowing that the covariance is zero does not prove that the random variables are independent. Independent variables have zero covariance, but the converse does not necessarily hold.

Answer 2

My usual trick for thinking about questions like this is to imagine that $Y = X$. Then $X$ and $Y$ are clearly dependent, for knowing $X$ gives me complete information about $Y$.

So in terms of tables of values: I have a table with two columns, namely $X$ and $Y$, and $n$ rows where $n$ is the number of samples.

• I cover the second column and calculate $E[X]$ and $E[X^2]$ from the first column.

• I cover the first column and calculate $E[Y]$ and $E[Y^2]$ from the first column.

• I notice that $E[X] = E[Y]$ and $E[X^2] = E[Y^2]$. This is intriguing but doesn’t yield any insight.

• I then go row by row and notice that they are values in each row are identical. This points to dependence.

In general, to establish independence (or the nature of the dependence), I’m looking for whether knowledge of the value in the first column can sharpen my guess about the value in the second column.