How to calculate mean of conditional probability?

Question

If I have $E[(A-B)|C]$ to calculate mean of something, is it equal with $E[A|C]-E[B|C]$?

If yes, where I can get the references? Or how to prove it?

Answer 1

Yes, conditional expectation is linear: consider the case for discrete variables $A$ and $B$:

$$E[c_1A + c_2B] = \sum_a \sum_b (c_1 a + c_2 b) P(A = a, B = b | C)$$

Expanding the product and using the linearity of sums:

$$c_1 \sum_a a \sum_b P(A = a, B = b | C) + c_2 \sum_a \sum_b b P(A = a, B = b | C)$$

By the law of total probability, $\sum_b P(A = a, B = b | C) = P(A = a | C),$ so we can simplify the first sum, and changing the order of summation in the second sum allows us to do the same there, yielding:

$$c_1 \sum_a a P(A = a|C) + c_2 \sum_b b P(B = b | C) = c_1 E[A | C] + c_2 E[B | C]$$