limit theorem

The central limit theorem states that any set of variates with a distribution having a finite mean and variance tends to the normal distribution. This allows statisticians to approximate sets of data with unknown distributions as being normal.

Let X_1, X_2, ..., X_N be a set of N independent random variates and each X_i have an arbitrary probability distribution P(x_1, ..., x_N) with mean μ_i and a finite variance σ_i^2. Then the normal form variate X_norm congruent ( sum_(i = 1)^N x_i - sum_(i = 1)^N μ_i)/sqrt( sum_(i = 1)^N σ_i^2) has a limiting cumulative distribution function which approaches a normal distribution. Under additional conditions on the distribution of the addend, the probability density itself is also normal with mean μ = 0 and variance σ^2 = 1.

Berry-Esséen theorem | Lindeberg condition | Lindeberg-Feller central limit theorem | Lyapunov condition

college level (AP statistics, California probability and statistics standard)