**AKA** Weak Law of Large Numbers. > [!NOTE] Theorem > Let $(\Omega,\mathcal{F},\mathbb{P})$ be a [[Probability Space|probability space]]. Let $(X_{i})_{i \geq 1}$ be [[Pairwise Independent Set of Discrete Real-Valued Random Variables|pairwise independent discrete real-valued random variables]] on $(\Omega,\mathcal{F},\mathbb{P})$ that are [[Square-Integrable Discrete Real-Valued Random Variable|square-integrable]] with the same [[Variance of Square-Integrable Discrete Real-Valued Random Variable|variance]] and [[Expectation of Discrete Real-Valued Random Variable|expectation]], $\mu$ and $\sigma^{2}$ respectively. Then for all $\varepsilon>0$ and $n \in \mathbb{N}$ $\mathbb{P}\left( \mu-\varepsilon \leq \frac{X_{1}+\dots+X_{n}}{n} \leq \mu+\varepsilon \right) \geq 1- \frac{\sigma^{2}}{\varepsilon^{2} n}$ **Proof**: Let $\overline{X}=\frac{X_{1}+\dots+X_{n}}{n}.$ By [[Square Root Law]], $\mathbb{E}[\overline{X}]= \mu$ and $\text{Var}(\overline{X})=\frac{\sigma^{2}}{n}.$ Let $\varepsilon>0.$ Using [[Chebyshev's Inequality for Square-Integrable Discrete Real-valued Random Variables]], $\begin{align} \mathbb{P}\left( \mu-\varepsilon \leq \frac{X_{1}+\dots+X_{n}}{n} \leq \mu+\varepsilon \right) &= 1 - \mathbb{P}\left( \left | \overline{X} - \mu \right | > \varepsilon \right) \\ &\geq 1 - \frac{\text{Var}(\overline{X})}{\varepsilon^{2}} \\ &= 1 - \frac{\sigma^{2}}{\varepsilon^{2} n }. \end{align}$ # Applications **Sampling**: The sequence of $(X_{i})_{i\geq 1}$ forms a random sample. $\overline{X}=\frac{1}{n}\sum_{i=1}^{n} X_{i}$ is the sample mean. The theorem asserts that the sample mean will converge in probability to the population mean, given by $\mathbb{E}[\overline{X}]=\mu,$ as $n\to \infty.$