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August 3rd, 2017, 06:05 PM   #1
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Probability: discrete or continuous

Probability distributions can exist in two cases -- discrete or continuous. Do they also exist as both?
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August 3rd, 2017, 06:18 PM   #2
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Probability distributions can exist in two cases -- discrete or continuous. Do they also exist as both?
Yes, those are called mixed.
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August 4th, 2017, 01:19 AM   #3
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For every mixed probability distribution is there another one which serves as its complement, such as its Fourier transform?

Do continuous distributions serve as complements to discrete distributions and vice versa?

Is a parameter of a continuous probability distribution a discrete wavelength, and a parameter of a discrete probability distribution a continuous wavelength?
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August 4th, 2017, 04:02 AM   #4
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Here's an answer from a physics perspective...

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For every mixed probability distribution is there another one which serves as its complement, such as its Fourier transform?
I'm not sure why you'd ever want to take the Fourier transform of a probability distribution (you normally take Fourier transforms of a stream of instances of something in order to derive a distribution, such as a number or spectral distribution), but there's a whole bunch of derived variables you can make from probability distributions, like expectation values, variance and actual distributions. I'd wager that most mixed probability distributions are just combinations of discrete and continuous distributions superimposed on each other and then renormalised, but I'm not aware of any specific phenomena where you'd want to do this. I suppose spectral analysis might make use of it (for example, if you wanted to calculate the probability of receiving a photon of a particular energy from some source).

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Do continuous distributions serve as complements to discrete distributions and vice versa?
In general, no, but I guess it would depend on the particular phenomenon you're investigating and whether the maths that you're applying requires it. I can't think of any applications that would require such a thing... You'd have to investigate it.

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Is a parameter of a continuous probability distribution a discrete wavelength, and a parameter of a discrete probability distribution a continuous wavelength?
The only way I can see to make "wavelength" relevant to probability distributions is to consider the wavelength to be the independent variable so you're looking at the probability of something as a function of wavelength. This happens, for example, with photon emission distributions from gases... you shine white light on a gas... the atoms in the gas will absorb photons of specific frequencies only and it will then emit photons of specific frequencies based on the energy level transitions of the orbital electrons (not necessarily the same as the absorbed ones. Therefore, if you look at the total emission of light from a gas, you'll get the same spectral distribution as the white light you originally shone on the gas punctuated with spikes and troughs associated with spectral lines (absorption and emission spectra in this case). I guess you could then normalise the whole received spectrum to have an area of 1, effectively creating a mixed probability distribution. In practise, spectral lines tend to get blurred so the distributions can be considered purely continuous, but the underlying phenomenon of photon emission has a discrete component for sure.

Just in case you're not aware... some gases can be described by a density as a function of position $\displaystyle \rho(x,y,z)$ and such distributions can be assigned a mean-free path (as a constant or variable) which uses the symbol $\displaystyle \lambda$. This shouldn't be confused with wavelength. The mean-free path is the typical distance that something travelling within the medium can travel before colliding with something. Low density gases have a higher mean-free path and vice versa, so the mean-free path is dependent on the distribution of particles in the gas.

Therefore, although wavelength is unlikely to be a relevant parameter for probability distributions, there are parameters that characterise actual distributions which are additional to the usual mean, variance, standard deviation and expectation variable. Therefore, although I can't think of any phenomena which use wavelength as a characteristic parameter of a distribution, it wouldn't shock me if you could find an application somewhere that does.
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