Joint Distribution Functions/ part6

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,

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.