= Bernoulli independence from zero covariance
{c}
{title2=$\operatorname{Cov}(X,Y)=ad-bc$}
For two <Bernoulli random variables> with joint probabilities $a,b,c,d$ in the order $(0,0),(0,1),(1,0),(1,1)$, their <covariance> is $d-(c+d)(b+d)=ad-bc$. Zero covariance fixes the $(1,1)$ probability to the product of its marginals; subtracting this from each marginal fixes all three remaining joint probabilities to their products. Thus <uncorrelated random variables> taking only zero and one are independent, including degenerate marginals. The implication is special to this two-point setting and fails for general distributions.
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