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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.
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:
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 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.
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.
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 ...
In probability theory, the central limit theorem (CLT) states that, in many situations, when independent and identically distributed random variables are added, their properly normalized sum tends toward a normal distribution. This article gives two illustrations of this theorem.
In probability theory, the central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample converges to a normal distribution as the sample size increases to infinity.