Actually, I was wondering if we say x,y ~ N(2;2), then can we say X=Y. Also, what would be the value of E[xy]. I need to calculate E[xy] and the only information given is x,y ~ N(2,2). Can we do it without assuming that x and y are independent?
Options are 2, 4, 8, 16.
No, you can not say that X=Y. That would mean the value of X is always equal to the value of Y. Consider this example : imagine that you pick two people in the population . X is the number of children of the first one and Y is the number of children of the second one. Both random variables have the same probability but are not the same. If you learn that X is 1, then you can not conclude that Y is 1 too.
In your case, X and Y have the same marginal probabilities but They can be : totally independent, lightly or strongly dependent or even be the same random variable (X=Y).
A system of 2 random variables is not entirely defined by the marginal probabilities of each variable ($N(2,2)$ in your case). To fully define it, you need the density of probability over (X,Y) : p(x,y) a function of two variables that integrates to 1. In your case, you have only the sum of this function on the 2 dimensions x and y. So you can not calculate $E[xy]=\int{p(x,y).xy.dx.dy}$