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In probability theory, conditional independence describes situations wherein an observation is irrelevant or redundant when evaluating the certainty of a hypothesis. . Conditional independence is usually formulated in terms of conditional probability, as a special case where the probability of the hypothesis given the uninformative observation is equal to the probability
A random variable X has a Bernoulli distribution if Pr(X = 1) = p and Pr(X = 0) = 1 − p for some p ∈ (0, 1).. De Finetti's theorem states that the probability distribution of any infinite exchangeable sequence of Bernoulli random variables is a "mixture" of the probability distributions of independent and identically distributed sequences of Bernoulli random variables.
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.
Conditional dependence of A and B given C is the logical negation of conditional independence (()). [6] In conditional independence two events (which may be dependent or not) become independent given the occurrence of a third event. [7]
Conditional probability; Conditioning (probability) Conditional expectation; Conditional probability distribution; Regular conditional probability; Disintegration theorem; Bayes' theorem; Rule of succession; Conditional independence; Conditional event algebra. Goodman–Nguyen–van Fraassen algebra
Independence of random variables in probability theory; Coprimality in number theory; The double tack up symbol (тлл, U+2AEB in Unicode [1]) is a binary relation symbol used to represent: Conditional independence of random variables in probability theory [2]
This rule allows one to express a joint probability in terms of only conditional probabilities. [4] The rule is notably used in the context of discrete stochastic processes and in applications, e.g. the study of Bayesian networks, which describe a probability distribution in terms of conditional probabilities.
Therefore, from Basu's theorem it follows that these statistics are independent conditional on , conditional on . This independence result can also be proven by Cochran's theorem . Further, this property (that the sample mean and sample variance of the normal distribution are independent) characterizes the normal distribution – no other ...