Moment Generating Function of Linear Transformation of Random Variable

Theorem
Let $X$ be a random variable.

Let $\alpha$ and $\beta$ be real numbers.

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

Let $M_X$ be the moment generating function of $X$.

Then the moment generating function of $Z$, $M_Z$, is given by:


 * $\map {M_Z} t = e^{\beta t} \map {M_X} {\alpha t}$