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How do I create a complex Gaussian joint PDF plot (example figure)?
You are confusing the "time domain" with the "frequency domain". The term "white noise" is based on an analogy with white light. The latter term means that all optical frequency components >>within the visible range<< are present with nominally equal intensity. When we refer to "white noise", again the meaning is context dependent. In the context of acoustics, it might mean that all frequency components in a band from 20 Hz to perhaps 15 kHz are present with equal spectral density, that frequency range being what human ears typically span. All of the above instances relate to the "frequency domain." But when we are looking at, for example, noise voltages, we may be interested in the "time domain" statistics. For example, what is the probability that the instantaneous noise voltage exceeds, say, 11.5 microvolts. In the time domain, "random" noise voltages tend to have a so-called "normal" or Gaussian amplitude distribution. This is especially likely to be true if the observed voltage is some kind of summation of >many< individual random electron motions. In that situation, a statistical concept called "the central limit theorem" becomes relevant. (can be googled!)
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The theorem states that for any sample, the probability of each of its individual components of the noise voltage to be outside that threshold is proportional to the square root of the sum of the squares of all of its other components. So, for example, if there are 10 million voltages and the median is 11.5 (as it very often is), the probability of an individual voltage exceeding 11.5 microvolts is then the sum of the square root of 10 million times the (10 million times median) average value. As you can see, this sort of quantization is generally beneficial for practical applications, and is a key ingredient of some of the best scientific codes in the world. This topic is a bit far-fetched for now, but the idea has been discussed here: I will add more on this in the days and weeks to come.