Markov's Inequality

Markov's inequality is used to check if \(\mathbf{X}\) is a non-negative random variable. If $\mathbf{X} \geq 0 ), then:

\[ P(\mathbf{X} \geq a) \leq \frac{E(\mathbf{X})}{a} \]

Let \(\mathbf{X}\) be any positive discrete random variable, so we can write:

\[ \begin{aligned} E(\mathbf{X}) &= \sum_x x P(\mathbf{X} = x) \\ &\geq \sum_{x \geq a} x P(\mathbf{X} = x) \\ &\geq \sum_{x \geq a} a P(\mathbf{X} = x) \\ &= a P(\mathbf{X} \geq a) \end{aligned} \]

Thus, Markov's inequality gives us the result:

\[ P(\mathbf{X} \geq a) \leq \frac{E(\mathbf{X})}{a} \]