Sums of Variances of Independent Trials

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Theorem

Let $\mathcal E_1, \mathcal E_2, \ldots, \mathcal E_n$ be a sequence of experiments whose outcomes are independent of each other.

Let $X_1, X_2, \ldots, X_n$ be discrete random variables on $\mathcal E_1, \mathcal E_2, \ldots, \mathcal E_n$ respectively.


Let $\operatorname{var} \left({X_j}\right)$ be the variance of $X_j$ for $j \in \left\{{1, 2, \ldots, n}\right\}$.


Then:

$\displaystyle \operatorname{var} \left({\sum_{j \mathop = 1}^n X_j}\right) = \sum_{j \mathop = 1}^n \operatorname{var} \left({X_j}\right)$


That is, the sum of the variances equals the variance of the sum.


Proof