# Moment generating Function: sample mean

Let $\bar{X_{n}}$ be the sample mean of a random sample of size n from a distribution which has a pdf of:

f(x) = ${e^{-x}}$, for x> 0; 0, other wise.

a) show that the mgf of $_{Y_{n}}=\sqrt{n}(\bar{X_{n}}-1)$ is $$\Psi _{Y_{n}}\left ( t \right )=\left ( e^{\frac{t}{\sqrt{n}}} - \frac{te^{\frac{t}{\sqrt{n}}}}{\sqrt{n}}\right)^{-n}$$ ,$t< \sqrt{n}$

So far I have been able to obtain:

$Y_{n}=\sqrt{n}\bar{X_{n}}-\sqrt{n}$, where a =$\sqrt{n}$ , b = $-\sqrt{n}$

Then

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