Why is the average of a sum equal the sum of the averages?

Question

I came across this website showing the proof of the above question in the Expected Value section. However, I do not quite understand why the probability of $xy$ becomes the probability of $x$ (or $y$) and why then the two summations reduce to only one in each term? Is it possible that $P_{xy}(x,y) = P_x(x) = P_y(y)$?

Answer 1

\begin{align} & \sum_x \sum_y x P_{XY}(x,y) = \sum_x \sum_y x \Pr(X=x\ \&\ Y=y) \\[10pt] = {} & \sum_x \left( x \sum_y \Pr(X=x\ \&\ Y=y) \right) \tag 1 \end{align} The justification of the last step above is that as (lower-case) $y$ runs through the list of all possible values of (capital) $Y$, the index $x$ does not change. It can therefore be pulled out. Then we can say \begin{align} & \sum_y \Pr(X=x\ \&\ Y=y) \\[10pt] = {} & \Pr\Big( (X=x\ \&\ Y=y_1) \text{ or }(X=x\ \&\ Y=y_2) \text{ or }(X=x\ \&\ Y=y_3)\text{ or }\cdots \Big) \\[10pt] = {} & \Pr\Big( (X=x) \ \&\ (Y=y_1\text{ or }Y=y_2\text{ or }Y=y_3\text{ or } \cdots)\Big) \\[10pt] = {} & \Pr(X=x) \\[10pt] = {} & P_X(x). \end{align} Consequently $(1)$ becomes $\displaystyle \sum_x x P_X(x)$.