Variance of Linear Combination of Random Variables/Corollary

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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.

$\blacksquare$