TITLE: Joint Bernoulli Distribution
QUESTION [5 upvotes]: If $X$ and $Y$ are two (not necessarily independent) Bernoulli's with success probabilities $a$ and $b$ resp., how do we construct the joint dist. in terms of $a$,$b$, and $\rho$---the correlation? 
I can get $\mathbb{P}(X=1,Y=1)$ by manipulating the expression for $\rho$, but lost for the other three...

REPLY [2 votes]: If you write $c:= P(X=1, Y=1)$, then you have, as others said, by definition of correlation coefficient:
$$
\rho = \frac{c - ab}{ \sqrt{a(1-a)b(1-b)} }
$$
and
$$
c = ab + \rho \sqrt{a(1-a)b(1-b)}
$$
Therefore the joint distribution is:
$P(X=1, Y=1) =c$
$P(X=1, Y=0) =a-c$
$P(X=0, Y=1) =b-c$
$P(X=0, Y=0) =1-a-b+c$
(it is simply filling in the contingency table).