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.

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- Aug 14th 2009, 04:47 AMfunnyingaCharacteristic Function of a symmetric RV
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.

- Aug 14th 2009, 08:05 AMmatheagle
If the distribution is symmmetric then all the odd moments are zero...

Then use

So

which no longer has those eye's in it, see my point? - Aug 15th 2009, 12:08 AMfunnyinga
Can you please explain how this is the case? I can't see it (Worried)

Quote:

Then use

So

which no longer has those eye's in it, see my point?

The characteristic function is , right? I can't see what you have done here.... - Aug 15th 2009, 12:24 AMmatheagle
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, .