Expectation calculation help? discrete random variables $X$ and $Y$.

Question

Is $E[X E[Y]] = E[X] E[Y]$? Even when $X$ and $Y$ are not independent? I am confused...

I thought that when they are independent, $E[XY] = E[X]E[Y]$. And then above sentence makes sense.

But does it still make sense when they are not independent?

Answer 1

For every random variable and real number $a$, $E[aX]=aE[X]$. Apply with $a=E[Y]$.