Markov's Inequality

Definition

Let $X$ be a non-negative Random Variable, and $E(X)<\infty$, then $\forall a>0,P(X\ge a)\le \frac{E(X)}{a}$ where $\frac{E(X)}{a}$ serves as an upper bound for the probability $P(X\ge a)$

Extended Version for Non-negative Functions

Let $u(X)$ be a non-negative Function of a Random Variable $X$, and $E(X)<\infty$, then $\forall a>0,P[u(X)\ge a]\le \frac{E[u(X)]}{a}$ where $\frac{E[u(X)]}{a}$ serves as an upper bound for the probability $P[u(X)\ge a]$