1. Characteristic functions

Given that E(e^(itx)) (the characteristic function of a Cauchy rv)=e^(-abs(t)), use this fact to find the pdf for X bar for all n.

I have no idea how to do this, and it was on the test. I got 4/25 points, and the professor doesn't give the correct answers.

I just started computing moments, and trying to integrate stuff, but he just wrote NO! on my paper. We have never done this in class or in the book, not making excuses, but I would just like to know how to do it in case it comes up again on the final.

Thank you.

2. $\phi_{\bar{X}}(t) = E(e^{it\frac{X_{1}+X_{2}+...+X_{n}}{n}})$

assuming independent observations

$= E(e^{it\frac{X_{1}}{n}})E(e^{it\frac{X_{2}}{n}}) \cdot \cdot \cdot E(e^{it\frac{X_{n}}{n}}) = \phi_{X} \Big(\frac{t}{n}\Big)^{n}$ since all n random variables follow Cauchy's distribution

$= (e^{-|\frac{t}{n}|})^{n} = e^{-|t|}$

so the sample mean follows the same Cauchy distribution