A FEW USEFUL CONTINUOUS PROBABILITY DISTRIBUTIONS

In the following sections we describe some continuous probability distributions. The main criterion of classification is theoretical vs. empirical distributions. The former class consists of distributions that are characterized by a very few parameters; indeed, they can also be labeled as parametric distributions. Theoretical distributions will never fit empirical data exactly, but they provide us with very useful tools, as they can be justified by some assumption about the underlying randomness. Furthermore, they have PDFs in analytical form, which may help us in finding analytical solutions to a wide set of problems. On the contrary an empirical distribution will, of course, fit observed data very well, but there is a hidden danger in doing so: We might overfit the distribution, obtaining a PDF or a CDF that does fit the peculiarities of the observed sample, but does not describe the properties of the population very well.

Empirical distribution will be the last example we cover here. First we consider the few main theoretical distributions, to provide the reader with the essential feeling for them. We start from the simplest case, the uniform distribution; then we consider the triangular and the beta distributions, which may be used as rough-cut models when little information is available on the underlying uncertainty. Then we describe the exponential and the normal distributions. They play a central role in probability theory because of their properties and because they can be used as building blocks to obtain other distributions. We defer the treatment of a few distributions obtained from the normal to Section 7.7.2, as they require some background on sums of random variables.