I've had this in an exam, and I must say I can't see how to prove it...
How to prove that it's a real-valued function ?
In general, if a distribution is symmetric (X has same distribution as -X), then its characteristic function is real. This is even an equivalent condition since we always have (for a real valued r.v.) for all .
That was easy...
Yeah, I know the part for the characteristic function, but it wasn't a probability exam (though I quickly tried to find a relationship with it, en vain), so I wasn't programmed for thinking about this
It would even have helped me for another question in the same exercise...