Central Limit Theorem

The Central Limit Theorem (CLT) states that for i.i.d. samples $X_1,\dots,X_n$ with finite mean $\mu$ and variance $\sigma^2 < \infty$, the standardized sample mean converges in distribution to a standard normal: $\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \xrightarrow{d} \mathcal{N}(0,1)$, at rate $O(1/\sqrt{n})$; the underlying samples need not be normal.

normal approximation for probabilities and quantiles of $\bar{X}$ when its exact distribution is intractable — the theoretical basis of Z-tests, CI formulas and Bootstrap; interviews typically probe “why $\sqrt{n}$”, the convergence rate and when it breaks.

the LLN only asserts $\bar{X} \to \mu$ and Chebyshev gives only a crude $O(1/n)$ bound; the CLT gives distribution-level precision: fluctuations cancel in the sum, so the standard error decays as $1/\sqrt{n}$ — halving the error costs 4x the samples, and going from $n=100$ to $n=10000$ shrinks the error only 10x.

State the CLT. What conditions are required (i.i.d., finite variance)?

Why does the standardized mean converge at rate $1/\sqrt{n}$? Explain via fluctuation cancellation.

What is the standard error of $\bar{X}$? How many more samples halve the error?

How does the CLT differ from and relate to Chebyshev's inequality and the LLN?

Why does the CLT fail for infinite variance (e.g. Cauchy)? What happens to the sample mean?