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What is the conditional expectation of Y?
The conditional expectation, E(X |Y = y), is a number depending on y. If Y has an influence on the value of X, then Y will have an influence on the average value of X. So, for example, we would expect E(X |Y = 2) to be different from E(X |Y = 3).
What is e y?
E(X|Y) is expected value of values of X given values of Y E(X|Y=y) is expected value of X given the value of Y is y.
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Which is the conditional expected value of Y?
For x ∈ S, the conditional expected value of Y given X = x ∈ S is simply the mean computed relative to the conditional distribution. So if Y has a discrete distribution then E(Y ∣ X = x) = ∑ y ∈ Tyh(y ∣ x), x ∈ S and if Y has a continuous distribution then E(Y ∣ X = x) = ∫Tyh(y ∣ x)dy, x ∈ S
Which is the unconditional expectation of a conditional?
The unconditional expectation of A is, but the expectation of A conditional on B = 1 (i.e., conditional on the die roll being 2, 3, or 5) is, and the expectation of A conditional on B = 0 (i.e., conditional on the die roll being 1, 4, or 6) is
How to use conditional expectation in probability theory?
CONDITIONAL EXPECTATION: L2¡THEORY. Definition 1. Let (›,F,P) be a probability space and let G be a ¾¡algebra contained in F. For any real random variable X 2 L2(›,F,P), define E(X jG) to be the orthogonal projection of X. onto the closed subspace L2(›,G,P).
How does the σ algebra affect conditional expectation?
The σ-algebra controls the “granularity” of the conditioning. A conditional expectation over a finer (larger) σ-algebra retains information about the probabilities of a larger class of events. A conditional expectation over a coarser (smaller) σ-algebra averages over more events.