The expected value of the sum is the sum of the expected values

$ E[X+Y] = E[X] + E[Y]

$ E\left[\sum_i X_i\right] = \sum_iE[X_i]

proof

$ E[X + Y] = \sum_x\sum_y (x + y) P(X=x, Y=y) ... Definition of expected value

$ = \sum_x\sum_y x P(X=x, Y=y) + \sum_x\sum_y y P(X=x, Y=y) ... Change the order of addition

$ = \sum_x x \sum_y P(X=x, Y=y) + \sum_y y \sum_x P(X=x, Y=y) ... Constant bracketing

$ = \sum_x x P(X=x) + \sum_y y P(Y=y) ... marginalization

$ = E[X] + E[Y] ... Definition of expected value

Expected number of times

The number of times is the sum

$ \#\{x\in X | f(x) \} = \sum_{x\in X} [f(x)]

https://gyazo.com/815c8001c4a4f32e92c5b96b004c4ef1

$ 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[[f(x, y)]]

This is not peripheral...

linearity of expectation

Change order of operations

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