I have an exercise where I want to calculate $E[X-Y]$ where $X,Y$ are discrete R.V.
Since the mean is defined with a sum I hope that $E[X-Y]=E[X]-E[Y]$ but I am having a hard time checking it.
$X,Y$ are discrete so $E[X]=\sum_{i=1}^N x_i p_i,\quad E[Y]=\sum_{j=1}^M y_i q_j$ where $p_{i},q_{j}>0$ and $\sum p_{i}=\sum q_i = 1$.
So $E[X]+E[Y]=\sum_{i=1}^N x_i p_i + \sum_{j=1}^M y_i q_j$ and $E[X+Y]=\sum_{i,j}(x_i+y_j)p'_{ij}.$ But I don't know what $p'_{ij}$ is or how to continue.
I would appreciate any help here!