Finding covariance given mean and variance of both X and Y

Question

Say the distribution of $X$ is known, and the expected value and variance of $Y$ is known. Don't assume independence. Is this information enough to give the covariance of $X$ and $Y$? I am only needing to know $\Bbb E[XY]$ but does the exact distribution of $Y$ have to be known?

Answer 1

Usually, $$ \begin{split} \mathrm{Cov}(X,Y) &= \mathbb{E}[(X-\mathbb{E}[X])(Y - \mathbb{E}[Y])] \\ &= \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y] \end{split} $$ So if you know the means of the distributions, knowing covariance or knowing $\mathbb{E}[XY]$ is equivalent, one info yields the other.

If you know nothing, calculating $$ \mathbb{E}[XY] = \iint_{\mathbb{R}^2} xyf(x,y)\ dA $$ requires knowing the joint pdf of $X,Y$, denoted above as $f(x,y)$...

Answer 2

You covariance could be any number in the range $[-\sqrt{\text{Var}(X)\text{Var}(Y)},\sqrt{\text{Var}(X)\text{Var}(Y)}];$ it must lie in that range.