The Weak Law of Large Numbers

Markov Inequality

If \(X\geq0\) and \(a>0\), then

\[\mathbf{P}(X\geq a)\leq\dfrac{\mathbb{E}[X]}{a}.\]

Chebyshev Inequality

\[\mathbf{P}\!\left(\lvert X-\mu\rvert\geq c\right)\leq\dfrac{\sigma^2}{c^2}\]

The Weak Law of Large Numbers

For \(\epsilon>0\),

\[\mathbf{P}\!\left(\lvert M_n-\mu\rvert\geq \epsilon\right)\rightarrow 0\qquad\text{as}\qquad n\rightarrow\infty.\]