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The law of large numbers and the central limit theorem

Why averages converge and why they are normally distributed.

The law of large numbers: sample means converge to the expectation. The central limit theorem: their fluctuations, scaled by √n, are normal. Picture it: the histogram of averages narrowing and turning bell-shaped as n grows. Think it: the CLT is a fixed-point statement. The normal is the distribution that sums of itself leave unchanged.

Worked example · limit of (1 + 1/n)^n as n -> oo

Calculus · Limits

Limit of (1 + 1/n)^n as n → oo

\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n}

Step by step

  1. \lim_{n \to \infty^+-} \left(1 + \frac{1}{n}\right)^{n}

    Try direct substitution first.

  2. \

    As x grows without bound, compare the fastest-growing terms (or divide top and bottom by the highest power).

  3. = e

    Take the limit.

Reveal the answer
e

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How to: The law of large numbers and the central limit theorem

  1. Try direct substitution first.
  2. As x grows without bound, compare the fastest-growing terms (or divide top and bottom by the highest power).
  3. Take the limit.

Questions people ask

What is the difference between probability and statistics?

Probability goes from a known model to what the data should look like; statistics goes from data back to the model. Probability theory is the deductive half.

What does the law of large numbers promise?

That the average of many independent samples converges to the expected value. It says nothing about any single trial.

More in Probability Theory

Sample spaces and the axiomsRandom variables and expectationThe common distributions