= Solution
The five specified masses sum to one, so there are no other possible outcomes. Symmetry gives $\mathbb E X=0$, while $\mathbb E Y=1/4$. The two terms with $Y=1$ have opposite $X$ values and equal masses, so
$$
\mathbb E(XY)=\frac18-\frac18=0,\qquad\boxed{\operatorname{Cov}(X,Y)=\mathbb E(XY)-\mathbb E X\,\mathbb E Y=0.}
$$
Nevertheless $\mathbb P(X=0,Y=1)=0$, whereas $\mathbb P(X=0)\mathbb P(Y=1)=(1/2)(1/4)=1/8$. Therefore \b[$X$ and $Y$ are not independent]. They are <uncorrelated random variables>, illustrating that zero <covariance> alone does not imply independence.
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