Find an example of two variables $X,Y$ whose values are in $\{0,1\}$ s.t. $\rho(X,Y)=\frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}=-1$.
My idea is the following: I have proved through the Cauchy-Schwarz-Inequality that $|\rho(X,Y)|\leq 1$. By the inequality, equality holds iff $X$ and $Y$ are "linearly dependent". Through this idea, I, unfortunately, have only found examples of $\rho(X,Y)=1$, but not -1. Could someone help me?