# Thread: Prove that it's real

1. ## Prove that it's real

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

2. Originally Posted by Moo
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 $\displaystyle t$.

3. Originally Posted by Laurent
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 !