A Chebyshev bound for a spread

Idea. The spread $X-Y$ has mean 0 and a small variance because the two variables move together. Chebyshev turns that variance into a bound.

Step 1: the variance. $$\mathrm{Var}(X-Y)=4+4-2\cdot \frac{3}{4}\cdot 2\cdot 2=2.$$

Step 2: Chebyshev. $$\mathbb{P}(|X-Y|\ge 2)\le \frac{2}{2^2}=\frac{1}{2}.$$

How good is it? $X-Y$ is normal with standard deviation $\sqrt{2}$, so the exact probability is $2\left(1-\Phi (\sqrt{2})\right)\approx 0.157$. Chebyshev holds for every distribution, which is why it is loose for a normal one.

The point. High correlation makes the spread much less risky than either leg: each variable has standard deviation 2, but their difference only $\sqrt{2}$. In general $\mathrm{Var}(X-Y)=2{\sigma }^2(1-\rho )$ for equal variances.