1. ## Characteristic functions

If I know that $\displaystyle \phi(t)$ is the characteristic function of a random variable $\displaystyle X$, how do I go about showing that variants on $\displaystyle \phi$ are characteristic functions of some random variables?

I've got a bunch of these to do. Here's an example: show that $\displaystyle |\phi(t)|^2$ is the characteristic function of some random variable.

2. I'd actually seen that, but in my probability course we haven't seen any of these results so I figured I should find another way...I got it now, though, and the rest by using the definition of a characteristic function (as the expectation of e^(itX), which I never grasped very well in the context of my course), super-basic properties of complex numbers (which I'm not very familiar with and is why it hadn't occurred to me previously) and trig functions, and Euler.

Got a little prodding from my instructor in his office hours, but I may have damaged my "reputation" more than the point total of my assignment increased my grade due to the apparent remedial nature of my difficulties (evidenced as well by the incredulous single-eyebrow-lifting going on in that office when I'm there).

3. Originally Posted by cribby
how do I go about showing that variants on $\displaystyle \phi$ are characteristic functions of some random variables?
The best way, when available, is to guess which random variable they are the characteristic functions of...

For instance, prove from the definition that $\displaystyle \phi(-t)$ is the characteristic function of $\displaystyle -X$, that $\displaystyle \phi(-t)=\overline{\phi(t)}$ (complex conjugate), and that $\displaystyle \phi(t)^2$ is the characteristic function of $\displaystyle X+Y$ where $\displaystyle Y$ is independent of $\displaystyle X$ and has same distribution.
Then give another thought at your initial question: which random variable has characteristic function $\displaystyle |\phi(t)|^2$?