Independent and identically distributed random variables in the context of Data mining

Statistical fluctuations are fluctuations in quantities derived from many identical random processes. They are fundamental and unavoidable. It can be proved that the relative fluctuations reduce as the square root of the number of identical processes.

Statistical fluctuations are responsible for many results of statistical mechanics and thermodynamics, including phenomena such as shot noise in electronics.

In information theory, Shannon's source coding theorem (or noiseless coding theorem) establishes the statistical limits to possible data compression for data whose source is an independent identically-distributed random variable, and the operational meaning of the Shannon entropy.

Named after Claude Shannon, the source coding theorem shows that, in the limit, as the length of a stream of independent and identically-distributed random variable (i.i.d.) data tends to infinity, it is impossible to compress such data such that the code rate (average number of bits per symbol) is less than the Shannon entropy of the source, without it being virtually certain that information will be lost. However it is possible to get the code rate arbitrarily close to the Shannon entropy, with negligible probability of loss.

In statistics, regression toward the mean (also called regression to the mean, reversion to the mean, and reversion to mediocrity) is the phenomenon where if one sample of a random variable is extreme, the next sampling of the same random variable is likely to be closer to its mean. Furthermore, when many random variables are sampled and the most extreme results are intentionally picked out, it refers to the fact that (in many cases) a second sampling of these picked-out variables will result in "less extreme" results, closer to the initial mean of all of the variables.

Mathematically, the strength of this "regression" effect is dependent on whether or not all of the random variables are drawn from the same distribution, or if there are genuine differences in the underlying distributions for each random variable. In the first case, the "regression" effect is statistically likely to occur, but in the second case, it may occur less strongly or not at all.

In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used with this or similar meanings in many scientific and technical disciplines, including physics, acoustical engineering, telecommunications, and statistical forecasting. White noise refers to a statistical model for signals and signal sources, not to any specific signal. White noise draws its name from white light, although light that appears white generally does not have a flat power spectral density over the visible band.

In discrete time, white noise is a discrete signal whose samples are regarded as a sequence of serially uncorrelated random variables with a mean of zero and a finite variance; a single realization of white noise is a random shock. In some contexts, it is also required that the samples be independent and have identical probability distribution (in other words independent and identically distributed random variables are the simplest representation of white noise). In particular, if each sample has a normal distribution with zero mean, the signal is said to be additive white Gaussian noise.

In the theory of probability, the Glivenko–Cantelli theorem (sometimes referred to as the fundamental theorem of statistics), named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, describes the asymptotic behaviour of the empirical distribution function as the number of independent and identically distributed observations grows. Specifically, the empirical distribution function converges uniformly to the true distribution function almost surely.

The uniform convergence of more general empirical measures becomes an important property of the Glivenko–Cantelli classes of functions or sets. The Glivenko–Cantelli classes arise in Vapnik–Chervonenkis theory, with applications to machine learning. Applications can be found in econometrics making use of M-estimators.