Let X be a random variable with mean and let E[(X ) 2k ] exist.

Let X be a random variable with mean μ and let E[(X − μ)^2k] exist. Show, with d > 0, that P(|X − μ| ≥ d) ≤ E[(X − μ)^2k]/d^2k. This is essentially Chebyshev’s inequality when k = 1. The fact that this holds for all k = 1, 2, 3, . . . , when those (2k)th moments exist, usually provides a much smaller upper bound for P(|X − μ| ≥ d) than does Chebyshev’s result.