For an integrable random variable and a sub-sigma-algebra , the conditional expectation is the almost-everywhere unique -measurable integrable random variable satisfyingfor every .
For nonnegative integrable , the finite measure restricted to a sub-sigma-algebra is absolutely continuous with respect to the restricted base measure. Its Radon-Nikodym derivative is -measurable and has the required integral identities. Positive and negative parts extend the construction to real integrable functions; real and imaginary parts extend it to complex functions. The result is unique almost everywhere.
If is a real square-integrable random variable and , then and the displayed identity follows by expansion. Thus equality of the second moments forces as an almost sure equality. This is the equality case of the conditional Jensen inequality for the strictly convex function .
Disintegrate over a factor map and take the norm using the conditional probability on each fiber. Its square integrates to the global squared norm. The uniform conditional L2 norm takes the essential supremum of these fiber norms.
For invertible probability systems linked by a factor map, invariance and uniqueness of disintegration identify the pushforward of each conditional measure with the conditional measure on the transformed fiber. The displayed norm identity follows for all integer times on a common full-measure set. It transports uniform conditional error bounds along whole orbits.
The essential supremum over the base of the conditional L2 norm. It may be infinite for a globally square-integrable function. For probability fibers, ; the uniform norm controls all fibers outside a null set rather than only their average.
For an integrable random variable , a sub-sigma-algebra , and a convex function with suitable integrability, almost surely. One proof writes as a supremum of supporting affine functions and applies monotonicity and linearity of conditional expectation to each. This proves the convex-loss form of the Rao-Blackwell theorem.
For an integrable random variable , the conditional expectation satisfies almost surely. Taking expected values gives the L1 contraction of conditional expectation. Applying it to shows that convergence in L1 is preserved by conditional expectation with respect to any fixed sigma-algebra.
For independent and identically distributed random variables that are integrable random variables, with , symmetry gives . The decreasing sigma-algebras generated by these future sums permit application of the reverse martingale convergence theorem.
Taking expectation after conditioning recovers the original expectation. More generally, if , then
If sub-sigma-algebras and are independent, every event in is independent of itself and consequently has probability zero or one. Thusalmost surely for every integrable random variable .
In general,Equality does hold when either sigma-algebra contains the other or when they are independent.
For nonnegative random variables and a sigma-algebra ,almost surely. It follows from the ordinary Fatou lemma after testing both sides on every event in .
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Conditional expectation is a fundamental concept in probability theory and statistics that refers to the expected value of a random variable given that certain conditions or information are known. It captures the idea of updating our expectations based on additional information. Formally, if \( X \) is a random variable and \( Y \) is another random variable (or an event), the conditional expectation of \( X \) given \( Y \) is denoted as \( \mathbb{E}[X | Y] \).