5. Prove the Chebyshev inequality is tight. In other words find a random variable X and a value \(a > 0\) such that \(P(|X - E[X]| > a) = \frac{Var(X)}{a^2}\). (Choose an example with \(Var(X) > 0\)).
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Step 1: To prove the Chebyshev inequality is tight, we need to find a random variable X and a value a>0 such that Var(X) P(|X-E[X]|>a) = a^2. Show more…
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Chebyshev's inequality states that, for λ > 0 and a random variable X, we have: P(|X - E X| ≥ λ) ≤ Var X / λ^2. For λ = 2, find an example of X which turns it into an equality, but with positive left- and right-hand sides.
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