Variance of Linear Transformation of Random Variable

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

$\blacksquare$