Joint probability distribution of two Bernoulli r.v. with a correlation $r$

Question

Suppose we have $X \sim Be(p_1)$, $Y \sim Be(p_2)$ and $corr(X,Y) = r$.

How does the joint distribution table look like?

For example, if $r = 0$ it is $$ \begin{array}{c|c|c} \cdot & T & F \\ \hline T & p_1p_2 & p_1(1-p_2) \\ \hline F & (1-p_1)p_2 & (1-p_1)(1-p_2) \\ \end{array} $$

Answer 1

Let $R=Cov(X,Y)= r*\sqrt{p_1(1-p_1)}*\sqrt{p_2(1-p_2)}$

Let $0\leqslant a,b,c,d \leqslant 1$ be the respective probabilities of TT, TF, FT and FF.

We need to solve the system:

$a+b=p_1$

$a+c=p_2$

$a+b+c+d=1$

$a=R+p_1p_2$

We get:

$a=R+p_1p_2$

$b=p_1(1-p_2)-R$

$c=p_2(1-p_1)-R$

$d=1+R-p_1-p_2+p_1p_2$

Thanks to @Henry for corrections.

Answer 2

Hint:

Note that here $\Pr(X=1\wedge Y=1)=\mathbb EXY$ and that the other $3$ probabilities can easily be deduced from this one.

So it is enough to find $\mathbb EXY$ and for this you can use the equality: $$\mathbb EXY-\mathbb EX\mathbb EY=\text{Cov}(X,Y)=\text{Corr}(X,Y)\sqrt{\text{Var}X\text{Var}Y}$$