Is there a closed form expression for noise other than white noise? For white noise, where the PSD is constant, we can model that as a simple Gaussian. Is there any way to model the probability density function of other types of noise (pink noise, red noise, blue noise, etc.), and if so, is there any way to derive such a thing?
This is usually done with filtering of white noise.


White noise generator + LP filter give you a red noise generator.
White noise generator + HP filter give you a blue noise generator


Design a filter with required frequency response and apply it to a white noise generator.