Calculate PMF of discrete random varialbe 
$p(x,y)=\frac{xy^2}{30}$ be the joint PMF of discrete random variables X and Y with the support $S=\{(x,y):x\in\{1,2,3\}, y\in\{1,2\}\}$

Compute $E(X-2Y)$
Question is can I expand to be $E(X)-2E(Y)$? (by linearity)?
 A: We compute the marginal PMF of $X$ by summing the joint PMF over the possible values of $Y$:
\begin{align}
\mathbb P(X=1) = \mathbb P(X=1,Y=1) + \mathbb P(X=1,Y=2) = \frac{1\cdot1^2}{30} + \frac{1\cdot 2^2}{30} = \frac16\\
\mathbb P(X=2) = \mathbb P(X=2,Y=1) + \mathbb P(X=2,Y=2) = \frac{2\cdot1^2}{30} + \frac{2\cdot 2^2}{30} = \frac13\\
\mathbb P(X=3) = \mathbb P(X=3,Y=1) + \mathbb P(X=3,Y=2) = \frac{3\cdot1^2}{30} + \frac{3\cdot 2^2}{30} = \frac12,
\end{align}
and similarly the marginal PMF of $Y$:
\begin{align}
\mathbb P(Y=1) &= \mathbb P(X=1,Y=1) + \mathbb P(X=2,Y=1) + \mathbb P(X=3,Y=1)\\
&= \frac{1\cdot 1^2}{30} + \frac{2\cdot1^2}{30} + \frac{3\cdot1^2}{30}\\
&= \frac 15\\
\mathbb P(Y=2) &=\mathbb P(X=1,Y=2) + \mathbb P(X=2,Y=2) + \mathbb P(X=3,Y=2)\\
&= \frac{1\cdot 2^2}{30} + \frac{2\cdot2^2}{30} + \frac{3\cdot2^2}{30}\\
&= \frac 45.
\end{align}
The expectation of $X$ is given by
\begin{align}
\mathbb E[X] &= 1\cdot\mathbb P(X=1) + 2\cdot\mathbb P(X=2) + 3\cdot\mathbb P(X=3) \\
&= 1\cdot\frac16+2\cdot\frac13+3\cdot\frac12\\
&= \frac73,
\end{align}
and the expectation of $Y$ by
\begin{align}
\mathbb E[Y] &= 1\cdot\mathbb P(Y=1) + 2\cdot\mathbb P(Y=2)\\
&= 1\cdot\frac15 + 2\cdot\frac45 = \frac95.
\end{align}
It follows from linearity of expectation that
$$
\mathbb E[X-2Y] = \mathbb E[X] - 2\mathbb E[Y] = \frac73 -2\cdot\frac95 = -\frac{19}{15}.
$$
