Английская Википедия:Elementary event

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Шаблон: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. 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>\{ k \},</math> where <math>k \in \N</math> if objects are being counted and the sample space is <math>S = \{ 1, 2, 3, \ldots \}</math> (the natural numbers).
• <math>\{ HH \}, \{ HT \}, \{ TH \}, \text{ and } \{ TT \}</math> if a coin is tossed twice. <math>S = \{ HH, HT, TH, TT \}</math> where <math>H</math> stands for heads and <math>T</math> for tails.
• All sets <math>\{ x \},</math> where <math>x</math> is a real number. Here <math>X</math> is a random variable with a normal distribution and <math>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.

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.

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>S</math> and not necessarily the full power set.

See also

• Шаблон:Annotated link
• Шаблон:Annotated link

Further reading

• Шаблон:Cite book
• Шаблон:Cite book