Central limit theorem | Wikipedia audio article
Channel: Subhajit Sahu
Date: 2018-12-27

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Central_limit_theorem


00:03:17 1 Independent sequences
00:03:28 1.1 Classical CLT
00:06:14 1.2 Lyapunov CLT
00:08:44 1.3 Lindeberg CLT
00:12:57 1.4 Multidimensional CLT
00:15:47 1.5 Generalized theorem
00:33:59 2 Dependent processes
00:34:46 2.1 CLT under weak dependence
00:34:56 2.2 Martingale difference CLT
00:36:05 3 Remarks
00:38:13 3.1 Proof of classical CLT
00:41:21 3.2 Convergence to the limit
00:41:27 3.3 Relation to the law of large numbers
00:41:37 3.4 Alternative statements of the theorem
00:44:20 3.4.1 Density functions
00:51:12 3.4.2 Characteristic functions
00:53:49 3.5 Calculating the variance
01:01:43 4 Extensions
01:01:54 4.1 Products of positive random variables
01:02:58 5 Beyond the classical framework
01:03:59 5.1 Convex body
01:11:07 5.2 Lacunary trigonometric series
01:11:16 5.3 Gaussian polytopes
01:12:31 5.4 Linear functions of orthogonal matrices
01:13:11 5.5 Subsequences
01:13:40 5.6 Random walk on a crystal lattice
01:15:28 6 Applications and examples
01:20:04 6.1 Simple example
01:20:08 6.2 Real applications
01:20:19 7 Regression
01:22:21 7.1 Other illustrations
01:23:53 8 History
01:24:56 9 See also
01:25:22 10 Notes
01:26:09 11 References
01:26:36 12 External links
01:26:47 Simple example
01:27:42 Real applications
01:29:33 Regression
01:30:30 Other illustrations
01:30:54 History
01:37:26 See also



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SUMMARY
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In probability theory, the central limit theorem (CLT) establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a "bell curve") even if the original variables themselves are not normally distributed.  The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.
For example, suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic mean of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the distribution of the average will be closely approximated by a normal distribution.  A simple example of this is that if one flips a coin many times the probability of getting a given number of heads in a series of flips will approach a normal curve, with mean equal to half the total number of flips in each series. (In the limit of an infinite number of flips, it will equal a normal curve.)
The central limit theorem has a number of variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations, given that they comply with certain conditions.
The earliest version of this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is now known as the de Moivre–Laplace theorem.
In more general usage, a central limit theorem is any of a set of weak-convergence theorems in probability theory. They all express the fact that a sum of many independent and identically distributed (i.i.d.) random variables, or alternatively, random variables with specific types of dependence, will tend to be distributed according to one of a small set of attractor distributions. When the variance of the i.i.d. variables is finite, the attractor distribution is the normal distribution. In contrast, the sum of a number of i.i.d. random variables with power law tail distributions decreasing as |x|−α − 1 where 0  α  2 (and the ...