Limit Theorems

What happens when sample size $n$ becomes very large?

Weak Law of Large Numbers (WLLN)

The sample mean $\bar{X}_n$ converges in probability to the true mean $\mu$ . $\bar{X}_n \xrightarrow{P} \mu$

Central Limit Theorem (CLT)

The specific shape of the distribution matters less as $n$ grows.

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The Fundamental Theorem

For large $n$ , the standardized sample mean converges to the Standard Normal Distribution: $\frac{\sqrt{n}(\bar{X} - \mu)}{\sigma} \xrightarrow{d} Z \sim N(0,1)$

This justifies using the Normal approximation for many statistical problems.

Derived Distributions

From Normal samples, we derive:

Chi-Square ( $\chi^2$ )
t-distribution
F-distribution