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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 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.
This means that if we test the null hypothesis that the center of a Gaussian scale mixture distribution is 0, say, then t n G (x) (x ≥ 0) is the infimum of all monotone nondecreasing functions u(x) ≥ 1/2, x ≥ 0 such that if the critical values of the test are u −1 (1 − α), then the significance level is at most α ≥ 1/2 for all ...
A typical finite-dimensional mixture model is a hierarchical model consisting of the following components: . N random variables that are observed, each distributed according to a mixture of K components, with the components belonging to the same parametric family of distributions (e.g., all normal, all Zipfian, etc.) but with different parameters
A sequence of random variables that are i.i.d, conditional on some underlying distributional form, is exchangeable. This follows directly from the structure of the joint probability distribution generated by the i.i.d. form. Mixtures of exchangeable sequences (in particular, sequences of i.i.d. variables) are exchangeable.
An important example of normal variance-mean mixtures is the generalised hyperbolic distribution in which the mixing distribution is the generalized inverse Gaussian distribution. The probability density function of a normal variance-mean mixture with mixing probability density g {\displaystyle g} is
In probability theory and statistics, a mixture is a probabilistic combination of two or more probability distributions. [1] The concept arises mostly in two contexts: 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 ...
Assuming that the distribution is a mixture of two normal distributions then the expectation-maximization algorithm may be used to determine the parameters. Several programmes are available for this including Cluster, [63] and the R package nor1mix. [64] Other distributions