Probability distribution explained

In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of possible outcomes for an experiment.^{[1] [2]} It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).^[3]

For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for, and 0.5 for (assuming that the coin is fair). More commonly, probability distributions are used to compare the relative occurrence of many different random values.

Probability distributions can be defined in different ways and for discrete or for continuous variables. Distributions with special properties or for especially important applications are given specific names.

Introduction

A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often represented in notation by

\Omega , is the set of all possible outcomes of a random phenomenon being observed. The sample space may be any set: a set of real numbers, a set of descriptive labels, a set of vectors, a set of arbitrary non-numerical values, etc. For example, the sample space of a coin flip could be

Notes and References

1. Book: Everitt, Brian . The Cambridge dictionary of statistics. 2006. Cambridge University Press. 978-0-511-24688-3 . 3rd. Cambridge, UK. 161828328.
2. Book: Ash, Robert B.. Basic probability theory. 2008. Dover Publications . 978-0-486-46628-6 . Dover . Mineola, N.Y. . 66–69. 190785258.
3. Book: Probability and statistics: the science of uncertainty. Evans . Michael . 2010. W.H. Freeman and Co. Rosenthal . Jeffrey S. . 978-1-4292-2462-8 . 2nd. New York. 38. 473463742.