On coloured Gaussian noise
Colored Gaussian noise is by definition a wide-sense-stationary (WSS) process; that is, it has constant mean (all the random variables constituting the process have the same mean) and its autocorrelation function $R_X(t_1, t_2) = E[X(t_1)X(t_2)]$ depends only on the difference $t_2-t_1$ of the arguments. It is conventional to use $\tau$ to denote the difference $t_2-t_1$, and abuse notation by writing $R_X(\tau)$ instead of the more prolix $R_X(0, \tau) = R_X(t,t+\tau)$ for the autocorrelation function. The power spectral density (PSD) of the process is then the Fourier transform of $R_X(\tau)$:
$$S_X(f) = \int_{-\infty}^\infty R_X(\tau)e^{-j2\pi ft} \,\mathrm dt.$$
The PSD is an even nonnegative function of $f$.

White noise is a zero-mean process for which $R_X(\tau) = K\delta(\tau)$ where $\delta(\cdot)$ is the Dirac delta or impulse and its PSD has constant value $K$ for $-\infty &lt; f &lt; \infty$. Colored noise is a zero-mean process whose PSD is not constant for all $f$. Colored Gaussian noise is a process in which all the random variables are zero-mean correlated (jointly) Gaussian random variables with random variables separated by time $\tau$ having covariance $R_X(\tau)$. Note that the variance of all the random variables is $\sigma^2 = R_X(0)$.  The PSD has the connection to the PDF that the PSD determines the variance of the random variables in question via the following corollary to the inverse Fourier transform formula:
$$\sigma^2 = R_X(0) = \int_{-\infty}^\infty S_X(f) \,\mathrm df.$$
Note that all the random variables constituting the process have the same (Gaussian) PDF (and so the same mean and same variance) and the variance is not a time-varying function due to the noise being colored.