My professor made a rather concise statement in class, which sums to this: E(Y|X=xi) = constant. E(Y|X )= variable. Could anyone help me understand how the expectation is calculated for the second case? I understand that for different values of xi, we'll have different values for the expectation. This is where my thoughts are all muddled up: E(Y|X)=[itex]\sum[/itex]i yi*P(Y=yi|X) = [itex]\sum[/itex]i yi * P(X|Y=yi)*P(Y=yi)/P(X). Could anyone explain the above computation, and how that is a variable? Also, it is my understanding that summing the probability P(Y=yi|X) over all values of Y won't be 1. Is this true?