The generalization of the preceding two variable case is the joint probability distribution of n \, discrete random variables X _ 1, X _ 2, \ dots, X _ n which is:
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The following joint probability distributions contain the population cell probabilities, along with the corresponding population odds ratio ( " OR " ) and population log odds ratio ( " LOR " ):
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Popescu and Rohrlich's maximum algebraic violation of the CHSH inequality can be reached by a non-signalling box, referred to as a standard PR box after these authors, with joint probability given by:
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In probability theory and statistics "'Chow Liu tree "'is an efficient method for constructing a second-order product approximation of a joint probability distribution, first described in a paper by.
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A point of caution, however : despite the similarity in appearance, is not a genuine joint probability distribution, because regions under it do not represent mutually exclusive states, as required in the first axiom.
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The joint probability distribution for a pair of random variables can be expressed in terms of their cumulative distribution function F ( x, y ) = P ( X \ le x, Y \ le y ).
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If these two tests were perfectly redundant, the results of both would be identical to the omnibus test; probabilistically speaking, the joint probability of rejecting both would be ? when the complete null hypothesis was true.
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The non-signalling requirement imposes further conditions on the joint probability, in that the probability of a particular output " a " or " b " should depend only on its associated input.
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The bivariate central limit theorem states that the joint probability distribution for \ overline { C _ 1 } and \ overline { S _ 1 } in the limit of a large number of samples is given by:
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The concept, due to Ronald Fisher, is equivalent to the statement that, conditional on the value of a sufficient statistic for a parameter, the joint probability distribution of the data does not depend on that parameter.
