Law of large numbers

In a statistical context, laws of large numbers imply that the average of a random sample from a large population is likely to be close to the mean of the whole population.

Related Topics:
Statistical - Average - Population - Mean

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In probability theory, several laws of large numbers say that the average of a sequence of random variables with a common distribution converges (in the senses given below) to their common expectation, in the limit as the size of the sequence goes to infinity. Various formulations of the law of large numbers, and their associated conditions, specify convergence in different ways.

Related Topics:
Probability theory - Sequence - Random variables - Distribution - Converges - Expectation - Limit

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

When the random variables have a finite variance, the central limit theorem extends our understanding of the convergence of their average by describing the distribution of the standardised difference between the sum of the random variables and the expectation of this sum. Regardless of the underlying distribution of the random variables, this standardised difference converges in distribution to a standard normal random variable.

Related Topics:
Variance - Central limit theorem - Converges in distribution - Standard normal random variable

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The phrase "law of large numbers" is also sometimes used to refer to the principle that the probability of any possible event (even an unlikely one) occurring at least once in a series increases with the number of events in the series. For example, the odds that you will win the lottery are very low; however, the odds that someone will win the lottery are quite good, provided that a large enough number of people purchased lottery tickets.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~