Basic Properties
Let (Ω, M, P) be a probability space, and let N be a σ-subalgebra of M.
- Conditioning with respect to N is linear on the space of integrable real random variables.
- More generally, for every integrable N–measurable random variable Y on Ω.
- for all B ∈ N and every integrable random variable X on Ω.
- Jensen's inequality holds: If ƒ is a convex function, then
- Conditioning is a contractive projection
-
- for any s ≥ 1.
Read more about this topic: Conditional Expectation
Famous quotes containing the words basic and/or properties:
“Insecurity, commonly regarded as a weakness in normal people, is the basic tool of the actor’s trade.”
—Miranda Richardson (b. 1958)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)