Suppose \(X\) and \(Y\) are random variables. The conditional expectation of \(X\) given \(Y\) is denoted by \(E[X|Y]\). Then \(E[E[X|Y]]\) equals
Answer
(C)
Explanation
\(E[X|Y]\) is itself a random variable: a function of \(Y\) that gives the average of \(X\) for each value \(Y\) can take. Averaging it over the distribution of \(Y\) recovers the overall average of \(X\). For discrete variables,
\(E[E[X|Y]] = \sum_{y} E[X|Y = y]\,P(Y = y) = \sum_{y}\sum_{x} x\,P(X = x, Y = y) = \sum_{x} x\,P(X = x) = E[X]\).
This is the law of total expectation (the tower property). Option A is still random, not a fixed number, and B and D fail in general: if \(Y\) is independent of \(X\), then \(E[X|Y] = E[X]\), so the result is \(E[X]\) whatever \(E[Y]\) is.
Answer: C