$E[X+Y] = E[X] + E[Y]$
$E\left[\sum_i X_i\right] = \sum_iE[X_i]$
proof
<ul>
<li>$E[X + Y] = \sum_x\sum_y (x + y) P(X=x, Y=y)$ ... <a href="/en/Definition%20of%20expected%20value">Definition of expected value</a>
</li>
<li>$= \sum_x\sum_y x P(X=x, Y=y) + \sum_x\sum_y y P(X=x, Y=y)$ ... <a href="/en/Change%20the%20order%20of%20addition">Change the order of addition</a>
</li>
<li>$= \sum_x x \sum_y P(X=x, Y=y) + \sum_y y \sum_x P(X=x, Y=y)$ ... Constant bracketing
</li>
<li>$= \sum_x x P(X=x) + \sum_y y P(Y=y)$ ... <a href="/en/marginalization">marginalization</a>
</li>
<li>$= E[X] + E[Y]$ ... <a href="/en/Definition%20of%20expected%20value">Definition of expected value</a>
</li>
<li><a href="/en/Expected%20number%20of%20times">Expected number of times</a>
<ul>
<li><a href="/en/The%20number%20of%20times%20is%20the%20sum">The number of times is the sum</a></li>
</ul>
</li>
<li>$#{x\in X | f(x) } = \sum_{x\in X} [f(x)]$
</li>
<li><img src="https://gyazo.com/815c8001c4a4f32e92c5b96b004c4ef1/thumb/1000" alt="image">
</li>
<li>$E_Y\left[\sum_{x\in X} [f(x, y)] \right] = \sum_{y\in Y}\sum_{x\in X} [f(x, y)]p(Y=y) = \sum_{x\in X} E_Y<a href="/en/f(x%2C%20y)">f(x, y)</a>$
</li>
<li>This is not peripheral...
</li>
<li><a href="/en/linearity%20of%20expectation">linearity of expectation</a>
</li>
<li><a href="/en/Change%20order%20of%20operations">Change order of operations</a>
</li>
</ul>
<hr>
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The expected value of the sum is the sum of the expected values