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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...

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3 Answers 3

up vote 1 down vote accepted

Hint: $P(X=1) = P(X=1,Y=0) + P(X=1,Y=1)$

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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).

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The correlation coefficient is equal to $$\frac{E(XY)-ab}{\sqrt{a(1-a)}\sqrt{b(1-b)}}.$$ If you know the correlation coefficient, and $a$ and $b$, then you know $E(XY)$.

But $E(XY)=\Pr(X=1 \cap Y=1)$. From this, using Robrt Israel's hint, you can calculate the rest of the $\Pr(X=i \cap Y=j)$.

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