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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 ...
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
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, 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 .
Probability theory or probability calculus is the branch of mathematics concerned with probability.Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms.
Existence and uniqueness of the needed conditional expectation is a consequence of the Radon–Nikodym theorem. This was formulated by Kolmogorov in 1933. Kolmogorov underlines the importance of conditional probability, writing, "I wish to call attention to ... the theory of conditional probabilities and conditional expectations". [17]
A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value.