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The conditional expectation of rainfall for an otherwise unspecified day known to be (conditional on being) in the month of March, is the average of daily rainfall over all 310 days of the ten–year period that fall in March. Similarly, the conditional expectation of rainfall conditional on days dated March 2 is the average of the rainfall ...
Conditional probabilities, conditional expectations, and conditional probability distributions are treated on three levels: discrete probabilities, probability density functions, and measure theory. Conditioning leads to a non-random result if the condition is completely specified; otherwise, if the condition is left random, the result of ...
In other words, a sufficient statistic T(X) for a parameter θ is a statistic such that the conditional probability of the data X, given T(X), does not depend on the parameter θ. A Rao–Blackwell estimator δ 1 (X) of an unobservable quantity θ is the conditional expected value E(δ(X) | T(X)) of some estimator δ(X) given a sufficient ...
In probability theory, regular conditional probability is a concept that formalizes the notion of conditioning on the outcome of a random variable. The resulting conditional probability distribution is a parametrized family of probability measures called a Markov kernel .
In mathematics, non-commutative conditional expectation is a generalization of the notion of conditional expectation in classical probability. The space of essentially bounded measurable functions on a σ {\displaystyle \sigma } -finite measure space ( X , μ ) {\displaystyle (X,\mu )} is the canonical example of a commutative von Neumann algebra .
In probability theory, the Doob–Dynkin lemma, named after Joseph L. Doob and Eugene Dynkin (also known as the factorization lemma), characterizes the situation when one random variable is a function of another by the inclusion of the -algebras generated by the random variables.
Conditional expectation; Expectation (epistemic) Expectile – related to expectations in a way analogous to that in which quantiles are related to medians; Law of total expectation – the expected value of the conditional expected value of X given Y is the same as the expected value of X; Median – indicated by in a drawing above
The conditional expectation of given , and the conditional variance may be understood as follows. Given any particular value y of the random variable Y , there is a conditional expectation E ( X ∣ Y = y ) {\displaystyle \operatorname {E} (X\mid Y=y)} given the event Y = y .