# Markov and Chebyshev inequalities

(30 minutes to learn)

## Summary

Markov's inequality and Chebyshev's inequality are tools for bounding the probability of a random variable taking on extreme values. While the bounds are weak, they apply under very general conditions. One use of Chebyshev's inequality is to prove the weak law of large numbers.

## Context

This concept has the prerequisites:

- random variables (The inequalities bound the probabilities of random variables taking on extreme values.)
- expectation and variance (The inequalities involve expectations and variances.)

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

## -Paid-

→ A First Course in Probability

An introductory probability textbook.

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

→ Probability and Statistics

An introductory textbook on probability theory and statistics.

Location:
Section 4.8, "The sample mean," from pages 229 to 231

## See also

- Chernoff bounds are another method for bounding the probability of extreme values.
- The inequality can be extended to higher-order moments .
- Chebyshev's bound can be used to prove the weak law of large numbers .