Search results
Results From The WOW.Com Content Network
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 ...
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.
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 .
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).
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\}} .
Here [] stands for the expectation conditioned to the σ-algebra . This general statement reduces to the previous ones when the topological vector space T is the real axis , and G {\displaystyle {\mathfrak {G}}} is the trivial σ -algebra {∅, Ω} (where ∅ is the empty set , and Ω is the sample space ).
Roughly speaking, the prefix "sub-" is consistent because the current observation X n is less than (or equal to) the conditional expectation E[X n +1 | X 1,...,X n]. Consequently, the current observation provides support from below the future conditional expectation, and the process tends to increase in future time.