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.
$\blacksquare$