Linear Transformation of Gaussian Random Variable

Theorem
Let $\mu$, $\alpha$ and $\beta$ be real numbers.

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:


 * $\alpha X + \beta \sim \Gaussian {\alpha \mu + \beta} {\alpha^2 \sigma^2}$

Proof
Let $Z = \alpha X + \beta$.

Let $M_Z$ be the moment generating function of $Z$.

We aim to show that:


 * $Z \sim \Gaussian {\alpha \mu + \beta} {\alpha^2 \sigma^2}$

By Moment Generating Function of Gaussian Distribution and Moment Generating Function is Unique, it is sufficient to show that:


 * $\displaystyle \map {M_Z} t = \map \exp {\paren {\alpha \mu + \beta} t + \frac 1 2 \alpha^2 \sigma^2 t^2}$

We also have, by Moment Generating Function of Gaussian Distribution, that the moment generating function of $X$, $M_X$, is given by:


 * $\displaystyle \map {M_X} t = \map \exp {\mu t + \frac 1 2 \sigma^2 t^2}$

We have: