Show that for any r.v.s X and Y , E(Y
Chapter 9, Problem 37(choose chapter or problem)
Show that for any r.v.s X and Y , E(Y |E(Y |X)) = E(Y |X). This has a nice intuitive interpretation if we think of E(Y |X) as the prediction we would make for Y based on X: given the prediction we would use for predicting Y from X, we no longer need to know X to predict Y we can just use the prediction we have! For example, letting E(Y |X) = g(X), if we observe g(X) = 7, then we may or may not know what X is (since g may not be one-to-one). But even without knowing X, we know that the prediction for Y based on X is 7. Hint: Use Adams law with extra conditioning.
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