Chebyshev Bound for Bivariate Normal Difference

Probability · Medium · Free problem
Let $(X, Y) \sim \text{BVN}(0, 0, \sigma^2, \sigma^2, \rho)$ -- a bivariate normal with zero means, equal marginal variances $\sigma^2$, and correlation $\rho$. Using Chebyshev's Inequality, find an upper bound on $P(|X - Y| \geq \sigma)$ when $\sigma^2 = 4$ and $\rho = 3/4$.

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