How can I find the probability distribution $f_{\overline{x}}(n, \mu)$ of the sample mean from a sample of size n drawn from an exponential population with mean $\mu$?
I have no idea where to begin on this. I know the same problem is easy with the normal distribution substituted for the exponential, since $$Z = \frac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$$
yields
$$ f_{\overline{x}} = \sqrt{\frac{\sigma}{2\pi n}} e^{\frac{(\overline{x}-\mu)^2}{2 {\sigma}^2}}+\mu $$
But I don't think there is any similar identity with the exponential distribution, relating the sample mean to a well known distribution. Any help is appreciated.