Simple question related to Correlation & Covariance

Question

$X$ and $Y$ are two random variables with $\Bbb E[X] = \Bbb E[Y] = 1$ and $\Bbb E[X^2] = \Bbb E[Y^2] = 2$. Which of the following is not possible:

1. $\Bbb E[XY] > 0$
2. $\Bbb E[XY] < 0$
3. $\Bbb E[XY] = 0$
4. $\Bbb E[XY] \le 2$

I reached the following conclusions:

• $\operatorname{Cov}(X,Y) = \Bbb E[XY] - \Bbb E[X]\Bbb E[Y] = \Bbb E[XY]-1$
• $\operatorname{Var}(X) = \Bbb E[X^2] - \Bbb E[X]^2 = \operatorname{Var}(Y) = 1$
• $\operatorname{Corr}(X,Y) = \operatorname{Cov}(X,Y)$
• I noticed that this value is the maximum value for both correlation and covariance since $$\operatorname{Corr}(X,Y)=\frac{\operatorname{Cov}(X,Y)}{\sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}}$$ and $\operatorname{Corr}(X,Y) = \Bbb E[XY]-1$ but I still did not reach a conclusion regarding $\Bbb E[XY]$.

Any help is appreciated!

Answer 1

Since the correlation $\rho=\Bbb E[XY]-1$ and $|\rho|\le1$, we must have $0\le\Bbb E[XY]\le2$ so the second option is not possible.