# For random variables X and Y, what is Var(X+Y) in terms of Var(X) and Var(Y)?

Oct 17, 2015

If $X$ and $Y$ are independent random variables, the following formula is correct:
$V a r \left(X + Y\right) = V a r \left(X\right) + V a r \left(Y\right)$

#### Explanation:

In short, a sum of variances of two independent random variables equals to a variance of their sum.

For random variables that are not independent, this property is not guaranteed.

A detailed description of variance and its properties with examples and rather rigorous proofs can be found at Unizor in the Probability part of the course in Random Variables item under topics Variance and Variance of Sum.