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

$E[X+Y]=E[X]+E[Y]$ $E\left[{\sum }_i{X}_i\right]={\sum }_{i}E[X_i]$

proof

• $E[X+Y]={\sum }_x{\sum }_y(x+y)P(X=x,Y=y)$ … Definition of expected value

• $={\sum }_x{\sum }_{y}xP(X=x,Y=y)+{\sum }_x{\sum }_{y}yP(X=x,Y=y)$ … 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

• $={\sum }_{x}xP(X=x)+{\sum }_{y}yP(Y=y)$ … marginalization

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

• Expected number of times

  • The number of times is the sum

• $\#\{x\in X|f(x)\}={\sum }_{x\in X}[f(x)]$

• 

• $E_Y\left[{\sum }_{x\in X}[f(x,y)]\right]={\sum }_{y\in Y}{\sum }_{x\in X}[f(x,y)]p(Y=y)={\sum }_{x\in X}{E}_Y[[f(x,y)]]$

• This is not peripheral…

• linearity of expectation

• Change order of operations

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