The central limit theorem

The central limit theorem

The central limit theorem states that for a set of independent and identically distributed random variables, $X_1$, $X_2$, $\dots$ the following holds: $$ \lim_{N \rightarrow \infty} P\left( \frac{S_n/N - \mu}{\sigma/\sqrt{N}} \le z \right) = \Phi(z) $$ where $N$ is the number of random variables and $S_n = X_1 + X_2 + \dots$. In the above expression $\mu=\mathbb{E}(X)$ and $\sigma^2=\textrm{var}(X)$ so obviously this expression cannot be used if your random variable has either $\mathbb{E}(X)=\infty$ or $\textrm{var}(X)=\infty$. The central limit theorem is used to interpret the results from a set of experiments. Each experiment generates a random variable $X_k$, and we assume that the random variables generated by each of these experiments have the same underlying probability distribution function (they are identical) and each of these random variables is independent (the outcome of one experiment does not affect the outcomes of other experiments).

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