Mixture distribution

mixturemixture densitydensity mixturemixed distributionmixing densitymixing probability densitymixture modelmixturesthe mixtures of densities
In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized.wikipedia
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Random variable

random variablesrandom variationrandom
In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized. A distinction needs to be made between a random variable whose distribution function or density is the sum of a set of components (i.e. a mixture distribution) and a random variable whose value is the sum of the values of two or more underlying random variables, in which case the distribution is given by the convolution operator.
Not all continuous random variables are absolutely continuous, for example a mixture distribution.

Mixture model

mixture modelsGaussian mixture modelGMM
Data analysis concerning statistical models involving mixture distributions is discussed under the title of mixture models, while the present article concentrates on simple probabilistic and statistical properties of mixture distributions and how these relate to properties of the underlying distributions.
Formally a mixture model corresponds to the mixture distribution that represents the probability distribution of observations in the overall population.

Probability distribution

distributioncontinuous probability distributiondiscrete probability distribution
In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized. In cases where each of the underlying random variables is continuous, the outcome variable will also be continuous and its probability density function is sometimes referred to as a mixture density. The cumulative distribution function (and the probability density function if it exists) can be expressed as a convex combination (i.e. a weighted sum, with non-negative weights that sum to 1) of other distribution functions and density functions. Note that the formulae here reduce to the case of a finite or infinite mixture if the density w is allowed to be a generalized function representing the "derivative" of the cumulative distribution function of a discrete distribution.
Such quantities can be modeled using a mixture distribution.

Compound probability distribution

compound distributioncompoundingmixture
More general cases (i.e. an uncountable set of component distributions), as well as the countable case, are treated under the title of compound distributions.
In probability and statistics, a compound probability distribution (also known as a mixture distribution or contagious distribution) is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with (some of) the parameters of that distribution themselves being random variables.

Robust statistics

robustbreakdown pointrobustness
Parametric statistics that assume no error often fail on such mixture densities – for example, statistics that assume normality often fail disastrously in the presence of even a few outliers – and instead one uses robust statistics.
by replacing estimators that are optimal under the assumption of a normal distribution with estimators that are optimal for, or at least derived for, other distributions: for example using the t-distribution with low degrees of freedom (high kurtosis; degrees of freedom between 4 and 6 have often been found to be useful in practice ) or with a mixture of two or more distributions.

Exponential distribution

exponentialexponentially distributedexponentially
The question of multimodality is simple for some cases, such as mixtures of exponential distributions: all such mixtures are unimodal.
Gamma mixture: If λ ~ Gamma(scale = k, shape = θ) and X ~ Exponential(rate = λ) then the marginal distribution of X is Lomax(scale = 1/k, shape = θ)

Mixture (probability)

mixture
mixture (probability)
A mixture defining a new probability distribution from some existing ones, as in a mixture distribution or a compound distribution. Here a major problem often is to derive the properties of the resulting distribution.

Expectation–maximization algorithm

expectation-maximization algorithmexpectation maximizationexpectation-maximization
expectation-maximization (EM) algorithm
mixture distribution

Probability

probabilisticprobabilitieschance
In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized.

Statistics

statisticalstatistical analysisstatistician
In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized.

Multivariate random variable

random vectorvectormultivariate
The underlying random variables may be random real numbers, or they may be random vectors (each having the same dimension), in which case the mixture distribution is a multivariate distribution.

Joint probability distribution

joint distributionjoint probabilitymultivariate distribution
The underlying random variables may be random real numbers, or they may be random vectors (each having the same dimension), in which case the mixture distribution is a multivariate distribution.

Probability density function

probability densitydensity functiondensity
In cases where each of the underlying random variables is continuous, the outcome variable will also be continuous and its probability density function is sometimes referred to as a mixture density. The cumulative distribution function (and the probability density function if it exists) can be expressed as a convex combination (i.e. a weighted sum, with non-negative weights that sum to 1) of other distribution functions and density functions.

Cumulative distribution function

distribution functionCDFcumulative probability distribution function
In cases where each of the underlying random variables is continuous, the outcome variable will also be continuous and its probability density function is sometimes referred to as a mixture density. The cumulative distribution function (and the probability density function if it exists) can be expressed as a convex combination (i.e. a weighted sum, with non-negative weights that sum to 1) of other distribution functions and density functions.

Convex combination

combinationsconvex combinationsconvex linear combination
In cases where each of the underlying random variables is continuous, the outcome variable will also be continuous and its probability density function is sometimes referred to as a mixture density. The cumulative distribution function (and the probability density function if it exists) can be expressed as a convex combination (i.e. a weighted sum, with non-negative weights that sum to 1) of other distribution functions and density functions.

Uncountable set

uncountableuncountably infiniteuncountably
More general cases (i.e. an uncountable set of component distributions), as well as the countable case, are treated under the title of compound distributions.

Convolution

convolvedconvolvingkernel
A distinction needs to be made between a random variable whose distribution function or density is the sum of a set of components (i.e. a mixture distribution) and a random variable whose value is the sum of the values of two or more underlying random variables, in which case the distribution is given by the convolution operator.

Multivariate normal distribution

multivariate normalbivariate normal distributionjointly normally distributed
As an example, the sum of two jointly normally distributed random variables, each with different means, will still have a normal distribution.

Statistical population

populationsubpopulationsubpopulations
Mixture distributions arise in many contexts in the literature and arise naturally where a statistical population contains two or more subpopulations.

Statistical model

modelprobabilistic modelstatistical modeling
Data analysis concerning statistical models involving mixture distributions is discussed under the title of mixture models, while the present article concentrates on simple probabilistic and statistical properties of mixture distributions and how these relate to properties of the underlying distributions.

Generalized function

generalized functionsgeneralised functionalgebra of generalized functions
Note that the formulae here reduce to the case of a finite or infinite mixture if the density w is allowed to be a generalized function representing the "derivative" of the cumulative distribution function of a discrete distribution.

Parametric family

parameterized familyparametrized familyfamily
The mixture components are often not arbitrary probability distributions, but instead are members of a parametric family (such as normal distributions), with different values for a parameter or parameters.

Linear combination

linear combinationslinearly combined(finite) left ''R''-linear combinations
A general linear combination of probability density functions is not necessarily a probability density, since it may be negative or it may integrate to something other than 1.

Skewness

skewedskewskewed distribution
These relations highlight the potential of mixture distributions to display non-trivial higher-order moments such as skewness and kurtosis (fat tails) and multi-modality, even in the absence of such features within the components themselves.