Mean Square value of stochastic process given the spectral density

Question

A stationary stochastic process have a spectral density of $$ S_{XX}(\omega) = 1 - \frac{|\omega|}{8 \pi}. $$

What is the mean square value of the process?

Answer 1

I think i fired it out.

There is a property of the spectral density function that says that, for an stationary process $$ E[X^2(t)] = \int_{-\infty}^{\infty}S_{XX}(\omega)\,d\omega $$

Since the spectral density function must be non-negative, $-8\pi \leq \omega \leq 8\pi$ the mean square is $$ E[X^2(t)] = \int_{-8\pi}^{8\pi}\left(1 - \frac{|\omega|}{8\pi}\right)\,d\omega = 8 \pi $$