Applications of the Central Limit Theorem

Use the central limit theorem to approximate probabilities for sample means and sums of iid random variables, including the normal approximations to the binomial and Poisson distributions.

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Introduction

The central limit theorem (CLT) lets us approximate probabilities about sums and averages of iid random variables using the normal distribution, even when the underlying distribution is far from normal.

If $X_1, X_2, \ldots, X_n$ are iid with mean $\mu$ and finite variance $\sigma^2,$ then for large $n,$ the sample mean is approximately normal:

\bar{X}_n = \frac{1}{n}\sum\limits_{i=1}^n X_i \;\approx\; N\!\left(\mu,\; \frac{\sigma^2}{n}\right).

Equivalently, the sum is approximately normal:

S_n = \sum\limits_{i=1}^n X_i \;\approx\; N(n\mu,\; n\sigma^2).

To turn this into a probability calculation, we standardize to obtain an approximately standard normal random variable:

Z = \dfrac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} = \dfrac{S_n - n\mu}{\sigma\sqrt{n}} \;\approx\; N(0,1).

For example, suppose $X_1, \ldots, X_{64}$ are iid with $\mu = 10$ and $\sigma^2 = 4.$ Then $\bar{X}_{64} \approx N\!\left(10,\, \tfrac{4}{64}\right) = N(10,\, 0.0625),$ so its standard error is $\sigma/\sqrt{n} = 2/8 = 0.25.$ Therefore

P(\bar{X}_{64} > 10.5) \approx P\!\left(Z > \dfrac{10.5 - 10}{0.25}\right) = P(Z > 2) \approx 0.0228.