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:

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

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

$\map {M_X} t = \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$

We have:

\(\ds \map {M_Z} t\)

\(=\)

\(\ds e^{\beta t} \map {M_X} {\alpha t}\)

Moment Generating Function of Linear Combination of Independent Random Variables

\(\ds \)

\(=\)

\(\ds e^{\beta t} \map \exp {\alpha \mu t + \frac 1 2 \sigma^2 \paren {\alpha t}^2}\)

\(\ds \)

\(=\)

\(\ds \map \exp {\paren {\alpha \mu + \beta} t + \frac 1 2 \sigma^2 \alpha^2 t^2}\)

Exponential of Sum

$\blacksquare$