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These bounds are not the tightest possible with general bivariates even when feasibility is guaranteed as shown in Boros et.al. [9] However, when the variables are pairwise independent (=), Ramachandra—Natarajan [10] showed that the Kounias-Hunter-Worsley [6] [7] [8] bound is tight by proving that the maximum probability of the union of ...
One approach to estimating the covariance matrix is to treat the estimation of each variance or pairwise covariance separately, and to use all the observations for which both variables have valid values. Assuming the missing data are missing at random this results in an estimate for the covariance matrix which is unbiased. However, for many ...
Throughout this article, boldfaced unsubscripted and are used to refer to random vectors, and Roman subscripted and are used to refer to scalar random variables.. If the entries in the column vector = (,, …,) are random variables, each with finite variance and expected value, then the covariance matrix is the matrix whose (,) entry is the covariance [1]: 177 ...
Correlation matrix — a symmetric n×n matrix, formed by the pairwise correlation coefficients of several random variables. Covariance matrix — a symmetric n×n matrix, formed by the pairwise covariances of several random variables. Sometimes called a dispersion matrix. Dispersion matrix — another name for a covariance matrix.
We want to bound the probability that any is a subset of . We will bound it using the expectation of the number of A ∈ S {\displaystyle A\in S} such that A ⊆ Γ p {\displaystyle A\subseteq \Gamma _{p}} , which we call λ {\displaystyle \lambda } , and a term from the pairwise probability of being in Γ p {\displaystyle \Gamma _{p}} , which ...
In probability theory, the joint probability distribution is the probability distribution of all possible pairs of outputs of two random variables that are defined on the same probability space. The joint distribution can just as well be considered for any given number of random variables.
In statistics, probability theory and information theory, pointwise mutual information (PMI), [1] or point mutual information, is a measure of association.It compares the probability of two events occurring together to what this probability would be if the events were independent.
Given a set of random variables = (), let (=) be the probability of a particular field configuration in —that is, (=) is the probability of finding that the random variables take on the particular value .