Hi,
I've had this in an exam, and I must say I can't see how to prove it...
$\displaystyle F(u)=\int_{\mathbb{R}} \exp\left(-\tfrac{x^2}{2}+iux\right) ~dx$
How to prove that it's a real-valued function ?
($\displaystyle u\in\mathbb{R}$)
Thanks
Hi,
I've had this in an exam, and I must say I can't see how to prove it...
$\displaystyle F(u)=\int_{\mathbb{R}} \exp\left(-\tfrac{x^2}{2}+iux\right) ~dx$
How to prove that it's a real-valued function ?
($\displaystyle u\in\mathbb{R}$)
Thanks
It's real if it equals its conjugate... In this case, symmetry $\displaystyle x\mapsto -x$ shows that it is the case.
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.) $\displaystyle \overline{\Phi_{X}(t)}=\Phi_{-X}(t)$ for all $\displaystyle t$.
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...
Thanks !