Conditional Expectation
Given that X = 1, the conditional expectation of the random variable Y is E ( Y | X = 1 ) = 0.3. More generally,
for x = 0, ..., 10. (In this example it appears to be a linear function, but in general it is nonlinear.) One may also treat the conditional expectation as a random variable, — a function of the random variable X, namely,
The expectation of this random variable is equal to the (unconditional) expectation of Y,
namely,
or simply
which is an instance of the law of total expectation E ( E ( Y | X ) ) = E ( Y ).
The random variable E(Y | X) is the best predictor of Y given X. That is, it minimizes the mean square error E ( Y - f(X) )2 on the class of all random variables of the form f(X). This class of random variables remains intact if X is replaced, say, with 2X. Thus, E ( Y | 2X ) = E ( Y | X ). It does not mean that E (Y | 2X ) = 0.3 × 2X; rather, E ( Y | 2X ) = 0.15 × 2X = 0.3 X. In particular, E (Y | 2X=2) = 0.3. More generally, E (Y | g(X)) = E ( Y | X ) for every function g that is one-to-one on the set of all possible values of X. The values of X are irrelevant; what matters is the partition (denote it αX)
of the sample space Ω into disjoint sets {X = xn}. (Here are all possible values of X.) Given an arbitrary partition α of Ω, one may define the random variable E ( Y | α ). Still, E ( E ( Y | α)) = E ( Y ).
Conditional probability may be treated as a special case of conditional expectation. Namely, P ( A | X ) = E ( Y | X ) if Y is the indicator of A. Therefore the conditional probability also depends on the partition αX generated by X rather than on X itself; P ( A | g(X) ) = P (A | X) = P (A | α), α = αX = αg(X).
On the other hand, conditioning on an event B is well-defined, provided that P (B) ≠ 0, irrespective of any partition that may contain B as one of several parts.
Read more about this topic: Conditioning (probability), Conditioning On The Discrete Level
Famous quotes containing the words conditional and/or expectation:
“Computer mediation seems to bathe action in a more conditional light: perhaps it happened; perhaps it didnt. Without the layered richness of direct sensory engagement, the symbolic medium seems thin, flat, and fragile.”
—Shoshana Zuboff (b. 1951)
“The expectation that every neurotic phenomenon can be cured may, I suspect, be derived from the laymans belief that the neuroses are something quite unnecessary which have no right whatever to exist. Whereas in fact they are severe, constitutionally fixed illnesses, which rarely restrict themselves to only a few attacks but persist as a rule over long periods throughout life.”
—Sigmund Freud (18561939)