For x is amplitude of anything the crest factor is

and

RMS(signal) = Stdev(signal) if the mean signal is 0. μ=0 => RMS=σ.

The crest factor of a noise source is the peak to average ratio of the noise. This is a measure of the quality of the noise distributions and one way to measure its Gaussian nature. Noise theoretically has an unbounded distribution so that it should have an infinite crest factor but the physical realization of the noise generator will limit the output excursions, via the amplifiers, diode junctions etc.

For a digital system where the maximum value is determined (and limited) by the number of bits, CF can be changed by changing the standard deviation σ. In a system where max value is infinite (if system can generate infinite as an output), CF is infinite (best) too.

Gaussian dist pdf function x axis (generatable noise values) is infinite. But it is limited to CF in digital systems.

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