# Motivation for statistical distributions

As statisticians, we come across many distributions under the banners "discrete" or "continuous", and "univariate" or "multivariate". But can anyone provide a good reason behind the existence and motivation for so many distributions? How do we get them? And what can a layman understand from it?

What is the logic behind the existence of distributions?

• There are many dedicated texts that give you the details. At the opposite extreme is this one paragraph. Data come in different forms (e.g. counts can only be zero or positive integers, some measurements can be positive only, some are less restricted). Also, data come in different shapes: experience with data shows that. So, statisticians and others have proposed many different models, some with mathematical derivations and underlying ideas about generating processes, and others just proposed more or less empirically as shapes that might be fair fits for at least some distributions. – Nick Cox Oct 15 '13 at 8:16

## 1 Answer

In many cases a distribution can be described as a result of some idealized experiment. For example if we flip a fair coin $n$ times the number of heads will follow a binomial distribution with parameters $n$ and .5. These idealized experiments are often used as models; they are used as simplified representation of how the data came to be. There are obviously many such models, and as a consequence many distributions. If you want the logic behind all distributions, then that will require a book of many volumes, e.g.:

N. L. Johnson, S. Kotz and N. Balakrishnan (2000). Continuous Multivariate Distributions, Vol. 1 (second edition), New York: Wiley & Sons.

N. L. Johnson, S. Kotz and N. Balakrishnan (1997). Discrete Multivariate Distributions. New York: John Wiley & Sons.

N. L. Johnson, S. Kotz and N. Balakrishnan (1995). Continuous Univariate Distributions, Vol. 2 (second edition), New York: John Wiley & Sons.

N. L. Johnson, S. Kotz and N. Balakrishnan (1994). Continuous Univariate Distributions, Vol. 1 (second edition), New York: John Wiley & Sons.

N. L. Johnson, A. W. Kemp and S. Kotz (1992). Univariate Discrete Distributions (second edition), New York: John Wiley & Sons.

A shorter list of distributions that is more suitable/affordable for owning yourself is:

Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). Statistical distributions. Wiley

• Sir, the answer is simpler when we talk about the binomial experiment. The above mentioned texts as we move on never provide the motivation for the distribution. I intend to ask this question on many distributions, individually in fact. – Vani Oct 18 '13 at 18:17