The conditional expectation of a random variable \(X\) given another random variable \(Y\) is a function \(f(Y) = E[X|Y]\) defined by \(f(y) = E[X|Y=y].\)

Example: Consider rolling two 6 sided dice. Let \(X\) be the value of the first die, and let \(Y\) be the total. Find \(E[Y|X].\)

Let \(X_2\) be the value of the second die. Then \(Y = X + X_2.\) The expected value of \(X_2\) is \[E[X_2] = \frac{1+2+3+4+5+6}{6} = 3.5\] To compute \(E[Y|X],\) first find the value at certain points. We will start with \(E[Y|X=1].\) \begin{align} E[Y|X=1] & = E[X+X_2|X=1] \\ & = E[X|X=1] + E[X_2|X=1] \\ & = 1 + 3.5 \\ & = 4.5 \end{align} At \(X = 2,\) we get \begin{align} E[Y|X=2] & = E[X+X_2|X=2] \\ & = E[X|X=2] + E[X_2|X=2] \\ & = 2 + 3.5 \\ & = 5.5 \end{align} In general, if \(x \in \{1,2,3,4,5,6\},\) \begin{align} E[Y|X=x] & = E[X+X_2|X=x] \\ & = E[X|X=x] + E[X_2|X=x] \\ & = x + 3.5 \end{align} So, \(E[Y|X] = X+3.5.\)