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The convergence to the normal distribution is monotonic, in the sense that the entropy of increases monotonically to that of the normal distribution. [23] The central limit theorem applies in particular to sums of independent and identically distributed discrete random variables.
Comparison of probability density functions, () for the sum of fair 6-sided dice to show their convergence to a normal distribution with increasing , in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).
Then according to the central limit theorem, the distribution of Z n approaches the normal N(0, 1 / 3 ) distribution. This convergence is shown in the picture: as n grows larger, the shape of the probability density function gets closer and closer to the Gaussian curve.
Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables, each of which clusters around a mean value.
Galton box A Galton box demonstrated. The Galton board, also known as the Galton box or quincunx or bean machine (or incorrectly Dalton board), is a device invented by Francis Galton [1] to demonstrate the central limit theorem, in particular that with sufficient sample size the binomial distribution approximates a normal distribution.
The normal distribution defines a family of stable distributions. By the classical central limit theorem the properly normed sum of a set of random variables, each with finite variance, will tend toward a normal distribution as the number of variables increases. Without the finite variance assumption, the limit may be a stable distribution that ...
By definition, a consistent estimator B converges in probability to its true value β, and often a central limit theorem can be applied to obtain asymptotic normality: (,),
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.