elementary event
{{redirect2|Basic outcome|Atomic event|atomic events in computer science|linearizability}}
{{Probability fundamentals}}
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.{{cite book|last=Wackerly|first=Denniss|author2=William Mendenhall|author3=Richard Scheaffer|title=Mathematical Statistics with Applications|year=2002 |publisher=Duxbury|isbn=0-534-37741-6}} 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 where if objects are being counted and the sample space is (the natural numbers).
- if a coin is tossed twice. where stands for heads and for tails.
- All sets where is a real number. Here is a random variable with a normal distribution and 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.
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.{{cite book|last=Kallenberg|first=Olav|title=Foundations of Modern Probability|edition=2nd|year=2002|page=9|url=https://books.google.com/books?id=L6fhXh13OyMC|publisher=Springer|location=New York|isbn=0-387-94957-7}}
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 and not necessarily the full power set.
See also
- {{annotated link|Atom (measure theory)}}
- {{annotated link|Pairwise independence|Pairwise independent events}}
References
{{reflist}}
Further reading
- {{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}}
Category:Experiment (probability theory)
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