Standard Gaussian Random Variable as Transformation of Gaussian Random Variable

Theorem

Let $\mu$ be a real number.

Let $\sigma$ be a positive real number.

Let $X \sim \Gaussian \mu {\sigma^2}$ where $\Gaussian \mu {\sigma^2}$ is the Gaussian distribution with parameters $\mu$ and $\sigma^2$.

Then:

$\dfrac {X - \mu} \sigma \sim \Gaussian 0 1$

where $\Gaussian 0 1$ is the standard Gaussian distribution.

Proof

\(\ds \frac {X - \mu} \sigma\)

\(=\)

\(\ds \frac 1 \sigma X - \frac \mu \sigma\)

\(\ds \)

\(\sim\)

\(\ds \Gaussian {\frac \mu \sigma - \frac \mu \sigma} {\paren {\frac 1 \sigma}^2 \sigma^2}\)

Linear Transformation of Gaussian Random Variable

\(\ds \)

\(=\)

\(\ds \Gaussian 0 1\)

$\blacksquare$