I'm trying to prove linearity of expectation for discrete case, i.e. $$\sum \limits_i E\left[X_i\right] = E \left[\sum \limits_i X_i \right]$$
Firstly, I introduce the following notation: $p_{ij} = P(X_i = j)$
If $X_i$ doesn't take value $j$, then $p_{ij} = 0$.
Then:
$\sum \limits_i E\left[X_i\right] = \sum \limits_i \sum \limits_j j \cdot p_{ij} = \sum \limits_j j \cdot \sum \limits_i p_{ij} = \sum \limits_j j \cdot p_j = E\left[\sum \limits_i X_i \right]$
Is this correct?