Central Limit Theorem

From Qwiki

Let $X_j\quad j\in \{1,\cdots N\}\,$ be independent, identically distributed random variables with zero mean and finite variance. Let $Y = \frac{1}{N}\sum_{j=1}^{N}X_j$ be the arithmetic mean of . Then in the limit $N\rightarrow\infty\,$ the random variable $Y\,$ has a Gaussian distribution $P(y) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp\left[-\frac{y^2}{2 \sigma^2}\right]$

The proof of this theorem is an exercise in Transformation of Random Variables.

Explanation from MathWorld: http://mathworld.wolfram.com/CentralLimitTheorem.html

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Categories: Concept | Probability