Theory 1
In many contexts it is useful to consider random variables that are summations of a large number of variables.
Summation formulas:
and Suppose
is a large sum of random variables: Then:
If
and are uncorrelated (e.g. if they are independent):
Extra - Derivation of variance of a sum
Using the definition:
In the last line we use the fact that
for the first term, and the symmetry property of covariance for the second term with the factor of 2.