> [!NOTE] Theorem (Markov's Inequality for Non-negative Discrete Real-valued Random Variables) > Let $(\Omega,\mathcal{F},\mathbb{P})$ be a [[Probability Space|probability space]]. Let $X$ be a non-negative [[Integrable Discrete Real-Valued Random Variable|integrable discrete real-valued random variable]] on $(\Omega,\mathcal{F},\mathbb{P}).$ For all $x>0,$ $\mathbb{P}(X>x) \leq \frac{\mathbb{E}[X]}{x}$where $\mathbb{E}[X]$ denotes the [[Expectation of Discrete Real-Valued Random Variable|expectation]] of $X.$ **Proof**: Let $x>0$. Define the DRV $Y:\Omega\to \mathbb{R}$ by $Y(\omega) = \begin{cases} x & X(\omega) \geq x \\ 0 & \text{otherwise} \end{cases}$ Note that for all $\omega\in \Omega,$ $Y(\omega) \leq X(\omega)/x$ because either $X(\omega)\geq x,$ in which case $\frac{X(\omega)}{x} \geq 1 =Y(\omega)$; or $X(\omega)<x,$ in which case $\frac{X(\omega)}{x}>0=Y(\omega).$ So by [[Expectation of Discrete Real-Valued Random Variable is Monotone]], $\mathbb{E}[Y] \leq \mathbb{E}[X]$which proves the result since $\mathbb{E}[Y]=x \cdot \mathbb{P}(X\geq x)$ as $Y$ is a DRV whose [[Probability Mass Function of Discrete Real-Valued Random Variable|PMF]] is given by $p_Y(x)=\mathbb{P}(X\geqslant x),\quad p_Y(0)=\mathbb{P}(X<x).$ # Applications **Consequences**: We gain [[Chebyshev's Inequality for Square-Integrable Discrete Real-valued Random Variables]] by setting $X=\mathbb{E}[Y-\mathbb{E}[Y]]$ and $x$ to $a^{2}$ which gives $\mathbb{P}(|Y-\mathbb{E}[Y]|\geq a) \leq \frac{\text{Var}(Y)}{a^{2}}.$