# Statements > [!NOTE] Theorem > Let $(X_{i})_{i \geq 1}$ be a sequence of [[Mutually Independent Set of Discrete Real-Valued Random Variables|mutually independent discrete real-valued random variables]] with the same [[Probability Distribution of Real-Valued Random Variable|distribution]]. Let $\mu$ and $\sigma^{2}$ be their [[Expectation of Discrete Real-Valued Random Variable|mean]] and [[Variance of Square-Integrable Discrete Real-Valued Random Variable|variance]] respectively (note that this will be the same for each $X_{i}$). Then for $a<b,$ $\mathbb{P}\left( \mu + \frac{\sigma}{\sqrt{ n }} a \leq \frac{X_{1}+\dots+X_{n}}{n} \leq \mu + \frac{\sigma}{\sqrt{ n }} b \right) \to \int_{a}^{b} \frac{1}{\sqrt{ 2\pi }} e^{-x^{2}/2} \, dx $as $n \to \infty,$ or equivalently $ \mathbb{P}\left(a \leqslant \frac{X_1+\cdots+X_n-n \cdot \mu}{\sigma \cdot \sqrt{n}} \leqslant b\right) \approx \int_a^b \frac{1}{\sqrt{2 \pi}} e^{-x^2 / 2} \mathrm{~d} x$ that is, **the distribution of sample mean can be approximated by a [[Normal Distribution|normal distribution]] with parameters $\mu$ and $\frac{\sigma^{2}}{n}.$** 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 converges almost surely to the population mean, given by $\mathbb{E}[\overline{X}]=\mu,$ as $n\to \infty$; unlike the weak law of large numbers for DRVs which only asserts that the sample mean converges in probability to the population mean. # Proofs **Proof**: Follows directly from # Applications