Bernoulli's theorem

The weak law of large numbers (cf. the strong law of large numbers) is a result in probability theory also known as Bernoulli's theorem. Let X_1, ..., X_n be a sequence of independent and identically distributed random variables, each having a mean 〈X_i 〉 = μ and standard deviation σ. Define a new variable X congruent (X_1 + ... + X_n)/n. Then, as n->∞, the sample mean 〈x〉 equals the population mean μ of each variable.

asymptotic equipartition property | central limit theorem | Chebyshev inequality | frivolous theorem of arithmetic | law of truly large numbers | strong law of large numbers