# weak law of large numbers

(45 minutes to learn)

## Summary

Roughly, the laws of large numbers state that if a random variable is sampled many times, the average of all the values approaches the expectation. In particular, the weak law states that the probability of the average of n trials differing by more than some value epsilon goes to zero as n goes to infinity. Unlike the strong law, it only requires that the variables be uncorrelated, not necessarily independent.

## Context

This concept has the prerequisites:

- independent random variables (The weak law of large numbers is a statement about independent random variables.)
- expectation and variance (The weak law of large numbers is about convergence to an expectation.)
- Markov and Chebyshev inequalities (Chebyshev's inequality is used in the proof.)

## Core resources (read/watch one of the following)

## -Paid-

→ Probability and Statistics

An introductory textbook on probability theory and statistics.

Location:
Section 4.8, "The sample mean," pages 229-235

Other notes:

- The theorem and proof extend to the case where the random variables are uncorrelated, not necessarily independent.

→ A First Course in Probability

An introductory probability textbook.

Location:
Section 8.2, "Chebyshev's inequality and the weak law of large numbers," pages 430-433

Other notes:

- The theorem and proof extend to the case where the random variables are uncorrelated, not necessarily independent.

## Supplemental resources (the following are optional, but you may find them useful)

## -Paid-

→ Mathematical Statistics and Data Analysis

An undergraduate statistics textbook.

Location:
Section 5.2, "The law of large numbers," pages 177-180

## See also

- The strong law of large numbers is a stronger form of this theorem, but which applies in more restricted cases.
- The central limit theorem characterizes the distribution of the average value more precisely.