Practice Using Chebyshev's Inequality
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Chebyshev's inequality states that for any random variable X, whose distribution is unknown, but the corresponding expected value (mean) is µ and variance is σ^2, and for any positive number k, P(|X - µ| ≥ kσ) ≤ 1/k^2. Compute P(|X - µ| ≥ kσ) for the values of k = 1, 2, and 3. Now assume that X   N(µ, σ^2); i.e., X actually follows a Normal distribution. In this case, are the actual probabilities (corresponding to X   N(µ, σ^2)) close to the Chebyshev bound of 1/k^2?
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