Elementary event: Difference between revisions
Corrected a mistake. Tags: Mobile edit Mobile web edit |
m Task 18 (cosmetic): eval 4 templates: del empty params (1×); |
||
| Line 24: | Line 24: | ||
==Further reading== |
==Further reading== |
||
*{{cite book |last=Pfeiffer |first=Paul E. |year=1978 |title=Concepts of Probability Theory |
*{{cite book |last=Pfeiffer |first=Paul E. |year=1978 |title=Concepts of Probability Theory |publisher=Dover |isbn=0-486-63677-1 |page=18 }} |
||
*{{cite book |last=Ramanathan |first=Ramu |title=Statistical Methods in Econometrics |location=San Diego |publisher=Academic Press |year=1993 |isbn=0-12-576830-3 |pages=7–9 }} |
*{{cite book |last=Ramanathan |first=Ramu |title=Statistical Methods in Econometrics |location=San Diego |publisher=Academic Press |year=1993 |isbn=0-12-576830-3 |pages=7–9 }} |
||
Revision as of 01:09, 17 December 2020
| Part of a series on statistics |
| Probability theory |
|---|
 |
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 corresponds to precisely one outcome.
The following are examples of elementary events:
- All sets {k}, where k ∈ N if objects are being counted and the sample space is S = {1, 2, 3, ...} (the natural numbers).
- {HH}, {HT}, {TH} and {TT} if a coin is tossed twice. S = {HH, HT, TH, TT}. H stands for heads and T for tails.
- All sets {x}, where x is a real number. Here X is a random variable with a normal distribution and S = (−∞, +∞). 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 S and not necessarily the full power set.
See also
References
- ^ Wackerly, Denniss; William Mendenhall; Richard Scheaffer. Mathematical Statistics with Applications. Duxbury. ISBN 0-534-37741-6.
- ^ Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 9. ISBN 0-387-94957-7.
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.
 | This probability-related article is a stub. You can help Wikipedia by expanding it. |
 | This statistics-related article is a stub. You can help Wikipedia by expanding it. |
