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An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".
Animated examples of the CLT; General Dynamic SOCR CLT Activity; Interactive Simulation of the Central Limit Theorem for Windows; The SOCR CLT activity provides hands-on demonstration of the theory and applications of this limit theorem. A music video demonstrating the central limit theorem with a Galton board by Carl McTague
In probability theory, Lindeberg's condition is a sufficient condition (and under certain conditions also a necessary condition) for the central limit theorem (CLT) to hold for a sequence of independent random variables.
The central limit theorem implies that those statistical parameters will have asymptotically normal distributions. The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
Central limit theorem — The triangle distribution often occurs as a result of adding two uniform random variables together. In other words, the triangle distribution is often (not always) the result of the first iteration of the central limit theorem summing process (i.e. =). In this sense, the triangle distribution can occasionally occur ...
The characteristic function approach is particularly useful in analysis of linear combinations of independent random variables: a classical proof of the Central Limit Theorem uses characteristic functions and Lévy's continuity theorem. Another important application is to the theory of the decomposability of random variables.
The means and variances of directional quantities are all finite, so that the central limit theorem may be applied to the particular case of directional statistics. [2] This article will deal only with unit vectors in 2-dimensional space (R 2) but the method described can be extended to the general case.
The central limit theorem can provide more detailed information about the behavior of than the law of large numbers. For example, we can approximately find a tail probability of M N {\displaystyle M_{N}} – the probability that M N {\displaystyle M_{N}} is greater than some value x {\displaystyle x} – for a fixed value of N {\displaystyle N} .