= Solution
The <probabilities> satisfy $a,b,c,d\ge0$ and $a+b+c+d=1$. Since the variables are indicators,
$$
\mathbb EX=c+d,\quad\mathbb EY=b+d,\quad\mathbb E[XY]=d.
$$
Their <covariance> is
$$
\operatorname{Cov}(X,Y)=d-(c+d)(b+d)=ad-bc.
$$
Thus the necessary and sufficient condition for <uncorrelated random variables> is
$$
\boxed{ad=bc.}
$$
This uses zero <covariance> as the definition. If one variable is constant, a normalized correlation coefficient is undefined, but zero <covariance> and this criterion remain meaningful.
Back to article page
