Probability density function via its characteristic help

[perukas]

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

So here is a problem i cannot solve...I can't reach to something satifactory
your ideas would be very helpful

Homework Statement

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?

Homework Equations

So I know that the characteristic function is:

Φ(ω)=∫e^jωx*f×(x)dx

The Attempt at a Solution

So we know that e^jωx= 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?

Thanks in advance,
sorry for my english again..
this is a very nice forum

 

[mark726]

Hello and welcome to the forum! Your English is perfectly fine, don't worry about it.

To solve this problem, we can use the fact that the characteristic function uniquely determines the probability distribution function. This means that if we know the characteristic function, we can find the probability distribution function. However, we need to make sure that the characteristic function is well-defined and that it satisfies certain properties.

First, we need to show that the characteristic function is continuous. This means that as ω approaches 0, Φ(ω) approaches 1. This is because as ω approaches 0, the integral of e^jωx becomes the integral of 1, which is just x. Since the probability distribution function must integrate to 1, the characteristic function must approach 1 as ω approaches 0.

Next, we need to show that the characteristic function is bounded. This means that for any value of ω, Φ(ω) must be finite. This is because the integral of e^jωx is always bounded, and the probability distribution function must integrate to 1.

Once we have shown that the characteristic function is continuous and bounded, we can use the inverse Fourier transform to find the probability distribution function. This is because the inverse Fourier transform is well-defined for continuous and bounded functions. So, yes, you can continue from where you left off and use the inverse Fourier transform to find the probability distribution function.

As for the case where α<0, we can still use the same method, but the resulting probability distribution function may not be valid for all values of x. This is because the probability density function is only defined for x≥α, so for α<0, we may need to adjust the resulting probability distribution function to account for this restriction.

I hope this helps and good luck on your problem!