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In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution. This holds even if the original variables themselves are not normally distributed. There are several versions of the CLT, each applying in the ...
Donsker's theorem states that the empirical distribution function, when properly normalized, converges weakly to a Brownian bridge—a continuous Gaussian process. This is significant as it assures that results analogous to the central limit theorem hold for empirical processes, thereby enabling asymptotic inference for a wide range of ...
The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example: The binomial distribution B ( n , p ) {\textstyle B(n,p)} is approximately normal with mean n p {\textstyle np} and variance n p ( 1 − p ) {\textstyle np(1-p)} for large n {\displaystyle n} and for p ...
This section illustrates the central limit theorem via an example for which the computation can be done quickly by hand on paper, unlike the more computing-intensive example of the previous section. Sum of all permutations of length 1 selected from the set of integers 1, 2, 3
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 gives only an asymptotic distribution. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.
Berry–Esséen theorem; De Moivre–Laplace theorem; Lyapunov's central limit theorem; Misconceptions about the normal distribution; Martingale central limit theorem; Infinite divisibility (probability) Method of moments (probability theory) Stability (probability) Stein's lemma; Characteristic function (probability theory) Lévy continuity ...
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 is not normal.