The random variables $X$ and $Y$ are both $Be(1/2) = Bin(1,1/2)$-distributed and have correlation coefficient $1/2$. Determine $P(X=Y)$.
I would really appreciate some hints on this, cause I'm completely lost. I'd wanna work with this using conditioning, but there is no joint distribution to work with here - or is there? I don't find any material on "recovering" a joint distribution function from the marginal ones. I would have guessed the joint distribution might be multinomial, but that can't be the case here since the correlation is not negative. On stackexchange it appears this question is closest to mine, but I can't quite understand the answers given there.
Hint: the correlation coefficient of $X,Y$ is $\frac{E[(X-1/2)(Y-1/2)]}{\sqrt{\operatorname{Var}(X) \operatorname{Var}(Y)}}$. Here the variances are both $1/4$, so $E[(X-1/2)(Y-1/2)]=\frac{1}{8}$. Try to compute the joint distribution from here.
A more detailed hint:
$$E[(X-1/2)(Y-1/2)]=E[XY]-1/2E[X]-1/2E[Y]+1/4=E[XY]-1/4=1/8.$$
Hence $E[XY]=3/8$. But $E[XY]=P(X=1 \text{ and } Y=1)$. This is one of the four numbers we need to compute to find the joint distribution. How can we find the other three?