# Probability density function via its characteristic help

1. May 5, 2012

### perukas

Hi there,
This is my first post...
and be kind on my english please...:)

So here is a problem i cannot solve...I cant reach to something satifactory

1. The problem statement, all variables and given/known data
The probability density function f×(x) of the random variable X is zero when x<α, α>0.
Its characteristic function is Φ(ω).
Prove than if you know only the real part the characteristic Φ(ω) it is possible to find f×(x).
Is that possible when α<0????

2. Relevant equations
So I know that the characteristic function is:

Φ(ω)=∫ejωx*f×(x)dx

3. The attempt at a solution

So we know that ejωx= cos(ωχ)+j*sin(ωx)

and Φ(ω)=∫[cos(ωx)+j*sin(ωx)]f×(x)dx

and the real part of the charcteristic is Re(Φ)=∫cos(ωx)*fx(x)dx...

and I now that fx(x) is the inverse Fourier transformation of the characteristic:
fx(x)=(1/2π)∫e-jωχ*Φ(ω)dω

and e-jωx=cos(ωx)-jsin(ωx)

so:
fx(x)=(1/2π)*∫cos(ωx)Φ(ω)dω - (1/2π)j∫sin(ωx)Φ(ω)dω
but we need to show that we can find the pdf via the Real part of its characteristic:

so (1/2π)∫cos(ωx)Φ(ω)dω=....

so can I go on from here???