expectation
Jos Conditional Figueroa-Lpez
Purdue University
Math 515 / Stat 540 Spring, 2008
Conditional expectation
Basic denitions and properties
Basic denitions
Intuition: E(Y |X ) is the best predictor of the random variable Y based
on the value of the random variable X .
Concretely: E(Y |X ) is the function of X , say f (X ), that is closest" to Y
on average; that is, min E (Y g(X )) = E (Y f (X )) .
g 2 2
For discrete or (abs.) continuous r.v.s, there are explicit formulas in term
of the joint density function p(x, y ) and the marginal pX (x):
Discrete: E(Y |X ) = f (X ), where
f (x) = E(Y |X...
CondExpct