# How to use mgf to find the distribution of a standardised normal.

We are given that: $$Z=\frac{Y-\mu}{\sigma}$$

We want to show that, if $$Y\sim N(\mu ,\sigma^2)$$, then $$Z$$ is a standard normal random variable using the uniqueness of moment generating functions.

The textbook gives:

$$m_Z (t) = E[e^{tZ}]=E[e^{(t/\sigma)(Y-\mu)}]=m_{Y-\mu}(t/\sigma )$$

I cannot figure out why the last equality is true:

$$E[e^{(e/\sigma)(Y-\mu)}]=m_{Y-\mu}(t/\sigma )$$

• Hint: By definition, what would $m_{Y-\mu}(t/\sigma)$ be? – Semiclassical Apr 8 at 15:13