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P(A|B) may or may not be equal to P(A), i.e., the unconditional probability or absolute probability of A. If P(A|B) = P(A), then events A and B are said to be independent: in such a case, knowledge about either event does not alter the likelihood of each other. P(A|B) (the conditional probability of A given B) typically differs from P(B|A).
Given two jointly distributed random variables and , the conditional probability distribution of given is the probability distribution of when is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value of as a parameter.
The DBAR problem, or the ¯-problem, is the problem of solving the differential equation ¯ (, ¯) = for the function (, ¯), where () is assumed to be known and = + is a complex number in a domain.
Given a discrete random variable X of ordered values to be encoded, let () be the probability for any x in X.Define a function ¯ = < + Algorithm: For each x in X, Let Z be the binary expansion of ¯ ().
Identity testing is the problem of determining whether a given multivariate polynomial is the 0-polynomial, the polynomial that ignores all its variables and always returns zero. The lemma states that evaluating a nonzero polynomial on inputs chosen randomly from a large-enough set is likely to find an input that produces a nonzero output.
The binomial distribution is the basis for the p-chart and requires the following assumptions: [2]: 267 The probability of nonconformity p is the same for each unit; Each unit is independent of its predecessors or successors; The inspection procedure is the same for each sample and is carried out consistently from sample to sample
where (,) is the cross entropy of Q relative to P and () is the entropy of P (which is the same as the cross-entropy of P with itself). The relative entropy D KL ( P ∥ Q ) {\displaystyle D_{\text{KL}}(P\parallel Q)} can be thought of geometrically as a statistical distance , a measure of how far the distribution Q is from the distribution P .
In complex geometry, the ¯ lemma (pronounced ddbar lemma) is a mathematical lemma about the de Rham cohomology class of a complex differential form.The ¯-lemma is a result of Hodge theory and the Kähler identities on a compact Kähler manifold.