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:

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

  • 1
    $\begingroup$ Hint: By definition, what would $m_{Y-\mu}(t/\sigma)$ be? $\endgroup$ – Semiclassical Apr 8 at 15:13

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