Elementary event

From HandWiki

In probability theory, an elementary event, also called an atomic event or sample point, is an event which contains only a single outcome in the sample space.[1] Using set theory terminology, an elementary event is a singleton. Elementary events and their corresponding outcomes are often written interchangeably for simplicity, as such an event corresponding to precisely one outcome.

The following are examples of elementary events:

  • All sets [math]\displaystyle{ \{ k \}, }[/math] where [math]\displaystyle{ k \in \N }[/math] if objects are being counted and the sample space is [math]\displaystyle{ S = \{ 1, 2, 3, \ldots \} }[/math] (the natural numbers).
  • [math]\displaystyle{ \{ HH \}, \{ HT \}, \{ TH \}, \text{ and } \{ TT \} }[/math] if a coin is tossed twice. [math]\displaystyle{ S = \{ HH, HT, TH, TT \} }[/math] where [math]\displaystyle{ H }[/math] stands for heads and [math]\displaystyle{ T }[/math] for tails.
  • All sets [math]\displaystyle{ \{ x \}, }[/math] where [math]\displaystyle{ x }[/math] is a real number. Here [math]\displaystyle{ X }[/math] is a random variable with a normal distribution and [math]\displaystyle{ S = (-\infty, + \infty). }[/math] This example shows that, because the probability of each elementary event is zero, the probabilities assigned to elementary events do not determine a continuous probability distribution.

Probability of an elementary event

Elementary events may occur with probabilities that are between zero and one (inclusively). In a discrete probability distribution whose sample space is finite, each elementary event is assigned a particular probability. In contrast, in a continuous distribution, individual elementary events must all have a probability of zero because there are infinitely many of them— then non-zero probabilities can only be assigned to non-elementary events.

Some "mixed" distributions contain both stretches of continuous elementary events and some discrete elementary events; the discrete elementary events in such distributions can be called atoms or atomic events and can have non-zero probabilities.[2]

Under the measure-theoretic definition of a probability space, the probability of an elementary event need not even be defined. In particular, the set of events on which probability is defined may be some σ-algebra on [math]\displaystyle{ S }[/math] and not necessarily the full power set.

See also

References

  1. Wackerly, Denniss; William Mendenhall; Richard Scheaffer. Mathematical Statistics with Applications. Duxbury. ISBN 0-534-37741-6. 
  2. Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 9. ISBN 0-387-94957-7. https://books.google.com/books/about/Foundations_of_Modern_Probability.html?id=L6fhXh13OyMC. 

Further reading

  • Pfeiffer, Paul E. (1978). Concepts of Probability Theory. Dover. p. 18. ISBN 0-486-63677-1. 
  • Ramanathan, Ramu (1993). Statistical Methods in Econometrics. San Diego: Academic Press. pp. 7–9. ISBN 0-12-576830-3.