How to define dependent random variables that satisfy certain conditions?

Question

I need to define to dependent random variables $X$ and $Y$ such that $\mathrm{E}(X) = 19$ and $\mathrm{E}(Y) = 160$ but $\mathrm{Cov}(X,Y) = 0$.

I was thinking about it quite a while and still didn't get the point of how to approach it.

Also, is it possible to do the same with independent variables such that $\mathrm{Cov}(X,Y) > 0$ ?

Thanks

Answer 1

Here's one of the simplest examples. First let $$ X = 19 + \begin{cases} \phantom{-}1 \\ \phantom{-}0 \\ -1 \end{cases} $$ each with probability $1/3$, then let $Y = 160 + X^2 - \operatorname{E}(X^2) = 160 + X^2 - \dfrac 2 3.$

The covariance between $X$ and $X^2$ is $0$, but clearly they are dependent.