Variance of Linear Combination of Random Variables/Corollary

Theorem
Let $X$ and $Y$ be independent random variables. Let the variances of $X$ and $Y$ be finite.

Let $a$ and $b$ be real numbers.

Then the variance of $a X + b Y$ is given by:


 * $\var {a X + b Y} = a^2 \, \var X + b^2 \, \var Y$

Proof
From Variance of Linear Combination of Random Variables, we have:


 * $\var {a X + b Y} = a^2 \, \var X + b^2 \, \var Y + 2 a b \, \cov {X, Y}$

where $\cov {X, Y}$ is the covariance of $X$ and $Y$.

From Covariance of Independent Random Variables is Zero:


 * $2 a b \, \cov {X, Y} = 0$

The result follows.