Binary random variables event-level independence implies random variable independence Can you help me in proving the following?
For two binary random variables X and Y, the event-level independence ($x^0 \perp y^0$) implies random variable independence $ X \perp Y$.

It is a matter of a system of equations type argument. Using the law of total probability, and the fact that the event $(X=0)$ is the complement of the event $(X=1)$ (same for $Y$), then we know:
$$ P(X=0) = P(X=0|Y=0)\cdot{}P(Y=0) + P(X=0|Y=1)\cdot{}P(Y=1)$$
$$ P(X=0) = P(X=0)\cdot{}P(Y=0) + P(X=0|Y=1)\cdot{}(1-P(Y=0))$$
Note that I used the known independence assumption to simplify the first term on the RHS going from line one to line two.
Now solve this expression for $P(X=0|Y=1)$
$$ P(X=0 | Y=1) = \frac{P(X=0) - P(X=0)P(Y=0)}{1-P(Y=0)} = P(X=0)$$
Thus, we now also know that $x^{0}$ is independent of $y^{1}$ too. Now, just repeat the same thing with $P(Y=1)$ with the new knowledge about $y^{1}$ and $x^{0}$ to get the result for $y^{1}$ and $x^{1}$.
