Chebyshev's Inequality

While Markov's Inequality only deals with means, Chebyshev's Inequality deals with both the mean and variance.

Let \(\mathbf{X}\), and define \(\mathbf{Y} = (\mathbf{X} - E(\mathbf{X}))^2\). Since \(\mathbf{Y}\) is a non-negative random variable, we apply Markov's inequality to \(\mathbf{Y}\). For any \(a > 0\), we have:

\[ \begin{aligned} E(\mathbf{Y}) &= (\mathbf{X} - E(\mathbf{X}))^2 = \text{Var}(\mathbf{X}) \\ P(|\mathbf{X} - E(\mathbf{X})| \geq a) &= P((\mathbf{X} - E(\mathbf{X}))^2 \geq a^2) \quad \text{let} \ a^2 = b \\ &= P(\mathbf{Y} \geq b) \leq \frac{E(\mathbf{Y})}{b} \\ P(|\mathbf{X} - E(\mathbf{X})| \geq a) &\leq \frac{\text{Var}(\mathbf{X})}{a^2} \end{aligned} \]

Chebyshev's Inequality bounds the probability that \(\mathbf{X}\) deviates from the mean by more than \(a\) standard deviations.