Markov's Inequality/Corollary

Corollary

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.

Then:

$\map \Pr {\size X \ge t} \le \dfrac {\expect {\size X} } t$

for each real $t > 0$.

Proof

From Markov's Inequality, we have:

$\ds \map \Pr {\set {\omega \in \Omega : \size {\map X \omega} \ge t} } \le \frac 1 t \int \size X \rd \Pr$

From the definition of expectation, we then have:

$\map \Pr {\set {\omega \in \Omega : \size {\map X \omega} \ge t} } \le \dfrac {\expect {\size X} } t$

So:

$\map \Pr {\size X \ge t} \le \dfrac {\expect {\size X} } t$

$\blacksquare$