Solve Fourier Transform Homework: Find Variance & Covariance

[peter.a]

Homework Statement

Just something I am working through and am a bit stuck on.

Homework Equations

I have taken the Fourier transform of an RC circuit which gives me :
Y(ω)=((X(ω))/(1+iωτ))
If i take the voltage across the circuit as white noise then i get:
Y(ω)=σ^²/2π/(1+iωτ))
How can i find the variance function and covariance of this Fourier transform

The Attempt at a Solution

I am not sure how to do this

 

[marcusl]

First, welcome to PF. Regarding your questions, why are you trying to find these? Y is a complex function, so what do you even mean by variance? Individual variances of real and imaginary parts? Of its power spectral density |Y|^2?

The variances of Real(Y) and Imag(Y) are the third and second moments of a function known in statistics as a Cauchy distribution, and these moments are undefined. |Y|^2 is the Cauchy distribution itself, and it has undefined first and second moments (mean and variance).

 

[peter.a]

Of the spectral density which is X(ω)/(1+(ωτ)^2)) i had incorectly stated it in the post.