# Tag Info

61

Are random errors necessarily gaussian? Errors are very often Gaussian, but not always. Here are some physical systems where random fluctuations (or "errors" if you're in a context with the thing that's varying constitutes an error) are not Gaussian: The distribution of times between clicks in a photodetector exposed to light is an exponential distribution....

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The reason is probably the central limit theorem: When you add lots of independent random variables, their sum will form a normal distribution, irrespective of their individual probability distributions. This makes normal distributions a pretty good guess if you do not have information about the origin of the error or if you have multiple sources of error. ...

11

Answers here have generally addressed the different question of whether empirical variables should be Gaussian, but 21joanna12 asked about experimental errors, which admit a completely different analysis. The best resource on that question I can recommend is Chapter 7 of Probability Theory: The Logic of Science by E T Jaynes. In short, there are good reasons ...

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Various answers appear here; I will add something that's not here yet. First, in order that random errors have expected value $0$ and be equally likely to be positive or negative, it is not necessary that their distribution be symmetric about $0.$ It's easy to find lots of counterexamples to that. Now suppose  Y_i = \alpha_0 + \alpha_1 x_{1,i} + \cdots + ...

6

There are many examples of physical phenomena that seem to be governed by non-Gaussian statistics. For instance, the Levy distribution arises in the multiple scattering of light in turbid media, where the photon path length follows this distribution. I think any time you have rare, but important events, you will see non-Gaussian statistics, such as with the ...

5

This is an open problem, and a special case of a more general problem, namely the question whether the entanglement of formation (i.e., the minimum average pure state entanglement needed to form a mixed state) for Gaussian states can always be achieved by a decomposition into Gaussian states. AFAIK the problem has only been settled (in the positive) for the ...

3

Gaussian distributions are often approximations that work well enough. On the plus side, their median, mean and modes are the same by symmetry and the algorithms to find the variance and all of the other salient details are easy enough for high school, undergraduate and less math oriented scholars. On the down-side, the domain of the Gaussian distribution ...

3

Besides the Gaussian or normal distribution, the Cauchy, or Lorentzian or Breit–Wigner distribution shows up frequently in physics and engineering phenomenon and measurements. While the Gaussian is its own Fourier transform, the Lorentzian is the Fourier transform of an exponential decay for t>0. Gaussian distirbution (solid line): $exp(-x^2)$ Cauchy ...

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First of all, Poisson distribution is a discrete distribution while Gaussian is continuous, so you can't really model a continuous noise using Poisson distribution and vice versa. The reason noise is usually modeled as a Gaussian random variable is largely due to Central Limit Theorem; since noise is typically result of many different effects acting on the ...

1

Quantization errors are a common practical example of a uniformly-distributed random error. For example, you have a digital scale which reads out to the nearest 0.1 gram. You put 2.5376 grams of powder in it and it reads "2.5". Then you on put 3.6264 grams of powder and it reads "3.6". And so on. Your readings have an error which, in this case, is a ...

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See also my answer here: Are negativity of the Wigner function and quantum behaviour equivalent? It seems that it is not clear whether the theorem can be extended to all kinds of mixed states. As you say, given a pure state, if we know that it is not Gaussian, Hudson's theorem implies that its Wigner function must be negative somewhere. For states not ...

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