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In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a subset of ...
The usual statement of the lemma is formulated in terms of one random variable being measurable with respect to the -algebra generated by the other. The lemma plays an important role in the conditional expectation in probability theory, where it allows replacement of the conditioning on a random variable by conditioning on the σ {\displaystyle ...
Thus, we postulate that the conditional expectation of given is a simple linear function of , {} = +, where the measurement is a random vector, is a matrix and is a vector. This can be seen as the first order Taylor approximation of E { x ∣ y } {\displaystyle \operatorname {E} \{x\mid y\}} .
In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra"; also σ-field, where the σ comes from the German "Summe" [1]) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair (,) is called a measurable space.
Then the unconditional probability that = is 3/6 = 1/2 (since there are six possible rolls of the dice, of which three are even), whereas the probability that = conditional on = is 1/3 (since there are three possible prime number rolls—2, 3, and 5—of which one is even).
Consider a Radon space (that is a probability measure defined on a Radon space endowed with the Borel sigma-algebra) and a real-valued random variable T. As discussed above, in this case there exists a regular conditional probability with respect to T .
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 .
These inequalities are significant for their nearly complete lack of conditional assumptions. For example, for any random variable with finite expectation, the Chebyshev inequality implies that there is at least a 75% probability of an outcome being within two standard deviations of the expected value. However, in special cases the Markov and ...