A random variable is called symmetric if and have the same distribution. Show that the characteristic function of a symmetric random variable takes values only in the reals.
A random variable is called symmetric if and have the same distribution. Show that the characteristic function of a symmetric random variable takes values only in the reals.
Calc 2, I obtained the Taylor Series for
I then let
, and so on.
If the distribution function is symmetric then the odd moments are 0, so those terms drop out of the series.
In the continuous case f(-x)=f(x) means that f(x) is an even function.
An even times an odd function is odd.
So, .