Elementary event: Difference between revisions
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{{Probability fundamentals}} |
{{Probability fundamentals}} |
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In [[probability theory]], an '''elementary event''' (also called an '''atomic event''' or '''simple event''') is an [[event (probability theory)|event]] which contains only a single outcome in the [[sample space]].<ref>{{cite book | last = Wackerly | first = Denniss | coauthors = William Mendenhall, Richard Scheaffer | title = Mathematical Statistics with Applications | publisher = Duxbury | isbn = 0-534-37741-6}}</ref> Using [[set theory]] terminology, an elementary event is a [[Singleton (mathematics)|singleton]]. Elementary events and |
In [[probability theory]], an '''elementary event''' (also called an '''atomic event''' or '''simple event''') is an [[event (probability theory)|event]] which contains only a single [[Outcome (probability)|outcome]] in the [[sample space]].<ref>{{cite book | last = Wackerly | first = Denniss | coauthors = William Mendenhall, Richard Scheaffer | title = Mathematical Statistics with Applications | publisher = Duxbury | isbn = 0-534-37741-6}}</ref> Using [[set theory]] terminology, an elementary event is a [[Singleton (mathematics)|singleton]]. Elementary events and their corresponding outcomes are often written interchangeably for simplicity, as such an event corresponds to precisely one outcome. |
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The following are examples of elementary events: |
The following are examples of elementary events: |
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* 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]]. |
* 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]]. |
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==Probability of an elementary event== |
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Elementary events may have probabilities that are strictly positive, zero, undefined, or any combination thereof. For instance, any [[discrete random variable|discrete]] probability distribution whose sample space is finite is determined by the [[Probability|probabilities]] it assigns to elementary events. In contrast, all elementary events have probability zero under any [[continuous random variable|continuous]] distribution. Mixed distributions, being neither entirely continuous nor entirely discrete, may contain ''atoms''. Atoms can be thought of as elementary (that is, ''atomic'') events with non-zero probabilities.<ref>{{cite book | last = Kallenberg | first = Olav | title = Foundations of Modern Probability | edition = 2nd | year = 2002 | publisher = Springer | location = New York | isbn = 0-387-94957-7}}</ref> Under the [[measure theory|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 [[sigma-algebra|σ-algebra]] on ''S'' and not necessarily the full [[power set]]. |
Elementary events may have probabilities that are strictly positive, zero, undefined, or any combination thereof. For instance, any [[discrete random variable|discrete]] probability distribution whose sample space is finite is determined by the [[Probability|probabilities]] it assigns to elementary events. In contrast, all elementary events have probability zero under any [[continuous random variable|continuous]] distribution. Mixed distributions, being neither entirely continuous nor entirely discrete, may contain ''atoms''. Atoms can be thought of as elementary (that is, ''atomic'') events with non-zero probabilities.<ref>{{cite book | last = Kallenberg | first = Olav | title = Foundations of Modern Probability | edition = 2nd | year = 2002 | publisher = Springer | location = New York | isbn = 0-387-94957-7}}</ref> Under the [[measure theory|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 [[sigma-algebra|σ-algebra]] on ''S'' and not necessarily the full [[power set]]. |
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Revision as of 18:20, 19 July 2013
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In probability theory, an elementary event (also called an atomic event or simple event) 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 = {0, 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 have probabilities that are strictly positive, zero, undefined, or any combination thereof. For instance, any discrete probability distribution whose sample space is finite is determined by the probabilities it assigns to elementary events. In contrast, all elementary events have probability zero under any continuous distribution. Mixed distributions, being neither entirely continuous nor entirely discrete, may contain atoms. Atoms can be thought of as elementary (that is, atomic) events with 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. Mathematical Statistics with Applications. Duxbury. ISBN 0-534-37741-6.
{{cite book}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - ^ Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. ISBN 0-387-94957-7.
- Pfeiffer, Paul E. (1978) Concepts of probability theory. Dover Publications. ISBN 978-0-486-63677-1 (online copy, p. 18, at Google Books)
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