# Central Limit Theorem

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The Central Limit Theorem identifies the distribution of the sample mean and is arguably the most important theorem in probability theory.

Let $X$ be a random variable, and let $X_1, X_2, \ldots , X_n$ be a random sample for $X$, such that each $X_i$ has a distribution identical to that of $X$ itself. Let $\overline{X}$ be the sample mean; in other words, let $\overline{X}$ be equal to $\frac{\sum X_i}{n}$. Because each $X_i$ is a random variable, $\overline{X}$ is also a random variable. The Central Limit Theorem observes several important facts about the distribution of $\overline{X}$:

1. The distribution of $\overline{X}$ is approximately normal, even when the underlying distribution $X$ is not.
2. The expected value of the $\overline{X}$ is equal to the expected value of $X$.
3. As the sample size $n$ increases, the variance of $\overline{X}$ approaches zero.