ENCYCLOPEDIA 4U .com

 Web Encyclopedia4u.com

# Law of large numbers

In probability theory, the weak law of large numbers states that if X1, X2, X3, ... is an infinite sequence of random variables, all of which have the same expected value μ and the same finite variance σ2, and they are uncorrelated (i.e., the correlation between any two of them is zero), then the sample average

converges in probability to μ. Somewhat less tersely: For any positive number ε, no matter how small, we have
Chebyshev's inequality is used to prove this result.

A consequence of the weak law of large numbers is the asymptotic equipartition property.

The strong law of large numbers states that if X1, X2, X3, ... is an infinite sequence of random variables that are independent and identically distributed, and have a common expected value μ then

i.e., the sample average converges almost surely to μ.

This law justifies the intuitive interpretation of the expected value of a random variable as the "long-term average when sampling repeatedly".

Content on this web site is provided for informational purposes only. We accept no responsibility for any loss, injury or inconvenience sustained by any person resulting from information published on this site. We encourage you to verify any critical information with the relevant authorities.