Is the conditional expectation $E[ f(X) \mid X]$ for a bounded deterministic function f on random variable X, always going to be $0$? Can someone please point out a reference or explain this?
2026-04-02 11:11:19.1775128279
Conditional expectation on functions of random variable
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No clearly not. We can easily compute that $$E[f(X) | X=x] = E[f(x) | X=x] = f(x)$$ and therefore $$E[f(X) \:|\: X] = f(X)$$ We do however have, that $$Var(f(X) \: | \: X) = E[f(X)^2 | X] - E[f(X)|X]^2 = f(X)^2-f(X)^2 = 0$$