I have the following probabilities: $$\begin{matrix}0.11&0.04&0.19\\0.19&0.12&0.19\\0.09&0.07&0\\\end{matrix}$$
With $x$ random variables horizontally, and $y$ random variables vertically. starting from $x = 0, 1, 2$ and $y = 0, 1, 2$.
I have to calculate $E(X|Y)$. I know that $p_{X|Y}(x|y) = \frac{P(X=x \space \cap \space Y=y)}{P(Y=y)}$ where $P(Y=y)$ is the marginal density for the discrete joint distribution.
We have the following: $$\begin{align} \sum_{i=0}^3P_{X|Y}(X=i|Y=0) &= \frac{0.11}{0.34}+\frac{0.04}{0.34} +\frac{0.19}{0.34} \\ \sum_{i=0}^3P_{X|Y}(X=i|Y=1) &= \frac{0.19}{0.50}+\frac{0.12}{0.50} +\frac{0.19}{0.50} \\ \sum_{i=0}^3P_{X|Y}(X=i|Y=2) &= \frac{0.09}{0.16}+\frac{0.07}{0.16} +\frac{0}{0.16}\end{align}$$
$$E(X|Y) = \sum_{i=0}^3x_iP_{X|Y}(X=i|Y=y) \\ = 0 \cdot \left(\frac{0.11}{0.34}+ \frac{0.19}{0.50} + \frac{0.09}{0.34} \right)+1 \cdot \left(\frac{0.04}{0.34}+ \frac{0.12}{0.50} + \frac{0.07}{0.16} \right)+2 \cdot \left(\frac{0.19}{0.34}+ \frac{0.19}{0.50} + \frac{0}{0.16} \right)=2.67$$
Are my calculations correct?
$E[X|Y]$ is a function of $Y$. \begin{align} E[X|Y=0] &= 0\cdot \frac{0.11}{0.34} + \frac{0.04}{0.34} + 2\cdot \frac{0.19}{0.34}\\ E[X|Y=1] &= 0\cdot \frac{0.19}{0.50}+ \frac{0.12}{0.50} + 2\cdot \frac{0.19}{0.50} \end{align}
I am leaving the evaluation of $E[X|Y=2]$ to you. If you insist you can use indicator variable to combine the formula.