Gaussian distribution random noise error

Question

1. Why do we in general consider the errors/noise in measured data are distributed in the Gaussian form? What is its advantage over the Laplace distribution?

2. Moreover, if we have to add some random noise to the data/model/fit, in order check its quality, why in general people opt for Gaussian random noise?

Thank you.

Praveen

Answer 1

1. Most of the errors in measured data can be approximated by the sum of a "large" number of independent random variables whose distributions are not known to us, but maybe we can find out their mean and variance. Then we can simply make use of Central Limit Theorem to infer that the resulting sum is Gaussian almost surely

   For example, in communication systems, the noise that we see in the receiver side, that gets added to the signal received is due to the independent, random fluctuations of the free electrons in the receiver circuit and the resulting noise is nothing but the sum of voltages induced by these random fluctuations. Then, by Central Limit Theorem, we can say that the resulting random noise is almost surely Gaussian with some mean and variance.

   The Laplace distribution comes into picture whenever we look for the difference between two exponential random variables. But, by CLT, most of the natural phenomenon are best modeled by Gaussian random variables than Laplace random variables. There are, however examples of Laplace random variables in use. In Fading channels, where the channel changes randomly with space as well as time due to shadowing and multipath effects, the square of the amplitude of the random channel gain is described quite well by an exponential random variable. There we see Laplace distribution coming into picture.

2. The use of Gaussian random variables as the first choice to describe some random measurement error is, i think, due to the CLT. Since most of the errors that happen during measurements are more or less independent of each other (though there are strong exceptions), CLT can kick in to give Gaussian distributions.

Edit: @user86927 In many practical situations, the measurement noise that you get is the sum of many independent identically distributed random variables, and in those cases the CLT kicks in. But the correct method to get the noise distribution should be first modeling the physical phenomenon behind the experiment and then proceed by identifying the different random variables involved and how they are contributing to the noise. If, for example, it happens that the resulting noise is actually a product of i.i.d. positive valued rv's then, in the limit the noise distribution is not gaussian, rather it is a log-normal distribution. So, ultimately it depends upon the model of the problem.