Find the covariance of two jointly distributed RV's when you don't know the joint pmf

Question

I have two correlated random variables, X and Y, and I am trying to find their covariance. I know: their individual expected values $\langle X \rangle$ and $\langle Y \rangle$ and their individual variances $\sigma^2_X$ and $\sigma^2_Y$. If the covariance is given as $cov(X,Y) = \langle XY \rangle -\langle X \rangle \langle Y \rangle$, then I already know the second term but I have no idea how to compute the first term. I don't know the joint pmf. Do I need the joint pmf to calculate $\langle XY \rangle$?

Answer 1

Consider two normal random variables $X$ and $Y$ each having mean zero and unit variance. If they are independent, then their covariance is $$\mathbb{E}[XY]-\mathbb{E}[X]\mathbb{E}[Y]=\mathbb{E}[X]\mathbb{E}[Y]-\mathbb{E}[X]\mathbb{E}[Y]=0.$$ However, if $X=Y$ (i.e., the two variables are perfectly correlated), then $$\mathbb{E}[XY]-\mathbb{E}[X]\mathbb{E}[Y]=\mathbb{E}[X^{2}]-\mathbb{E}[X]\mathbb{E}[Y]=1.$$

What does this suggest?