Variance of Linear Transformation of Random Variable
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Theorem
Let $X$ be a random variable.
Let $a, b$ be real numbers.
Then we have:
- $\var {a X + b} = a^2 \var X$
where $\var X$ denotes the variance of $X$.
Proof
We have:
\(\ds \var {a X + b}\) | \(=\) | \(\ds \expect {\paren {a X + b - \expect {a X + b} }^2}\) | Definition of Variance | |||||||||||
\(\ds \) | \(=\) | \(\ds \expect {\paren {a X + b - a \expect X - b}^2}\) | Expectation of Linear Transformation of Random Variable | |||||||||||
\(\ds \) | \(=\) | \(\ds \expect {a^2 \paren {X - \expect X}^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a^2 \expect {\paren {X - \expect X}^2}\) | Expectation of Linear Transformation of Random Variable | |||||||||||
\(\ds \) | \(=\) | \(\ds a^2 \var X\) | Definition of Variance |
$\blacksquare$