Find distribution of discrete random variables $X$ and $Y$ from conditional distributions $X|Y$ and $Y|X$

Question

Assume the two random variables X and Y that take as values either 0 or 1. Suppose I only know the following probabilities:

$p^X_{11}=P(X=1|Y=1)$

$p^X_{10}=P(X=1|Y=0)$

$p^Y_{11}=P(Y=1|X=1)$

$p^Y_{10}=P(Y=1|X=0)$

I want to find $p^X=P(X=1)$ and $p^Y=P(Y=1)$. So here is what I tried. I know the following holds:

$p^X = p^X_{11}*p^Y+p^X_{10}*(1-p^Y)$

$p^Y = p^Y_{11}*p^X+p^Y_{10}*(1-p^X)$

From there, I work with this system of equations and find:

$p^X = \frac{p^X_{10}+p^Y_{10}(p^X_{11}-p^X_{10})}{1-(p^X_{11}-p^X_{10})(p^Y_{11}-p^Y_{10})}$

$p^Y = \frac{p^Y_{10}+p^X_{10}(p^Y_{11}-p^Y_{10})}{1-(p^X_{11}-p^X_{10})(p^Y_{11}-p^Y_{10})}$

Is my reasoning correct? Thanks!

Answer 1

Yes.

Your starting equations are correct, and that is the solution to those simultaneous equations in two unknowns.