Let $ X(t) $ be a stationary random process with mean $ \mu_X = 1 $ and auto-covariance function $C_X = \begin{cases} 1 - |\tau| & -1 < |\tau| < 1 \\ 0 & otherwise \end{cases}$
How can you use the information given to calculate $ E[\lvert X(t)\rvert ^2]$?
So far, I have been using the following definition:
$ E[\lvert X(t)\rvert ^2] = R_{XX}(0) = \frac{1}{2\pi} \int_{-\infty}^{\infty} S_{XX}(j\omega) d\omega$, where $S_{XX}$ is the continuous time Fourier transform of $R_{XX}$, the autocorrelation function.
However, since I have the auto-covariance $C_{XX}$ already, I can get the autocorrelation function using the definition $C_{XX}(\tau) = R_{XX}(\tau) - \mu_{X}^2$. Then, I can simply evaluate $R_{XX}(0) = 2 - |0| = E[\lvert X(t)\rvert ^2] = 2$.
Is this correct? Or am I misusing a definition somewhere?