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I've been struggling with expected value calculations that involve multiple or nested sigma notations. For instance, the solution to a problem I was working on is:

$X=\Sigma_{i=1}^{10}X_i\implies E[X]=E(\Sigma_{i=1}^{10}X_i)=\Sigma_{i=1}^{10}E[X_i]$

Similarly,

$var(X)=var(\Sigma_{i=1}^{10}X_i)=\Sigma_{i=1}^{10}var(X_i)$

In both of these cases, I understand why the expected value and variance are being calculated as such, and I understand how the expected value and variance formulas can be expressed in terms of sigma notation, but I don't understand how these sigma notations are able to be swapped or brought inside/outside each other. I feel like I must have missed some fundamental algebraic property of sigma notation that allows for this maneuver.

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    $\begingroup$ The first is linearity of expectation and more generally $\mathbb E[aX+bY] =a\mathbb E[X]+b\mathbb E[Y]$. The second requires the $X_i$ to be independent (or at least have zero covariances) and more generally $\text{Var}(aX+bY) =a^2\text{Var}(X)+b^2\text{Var}(Y) +2ab \text{Cov}(X,Y)$. $\endgroup$
    – Henry
    Commented Aug 5, 2023 at 17:59

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