Random Experiment Event

AKA: E, Event.
Context:
- It can be instantiated in a Random Experiment Event Instance (with a random experiment outcome).
- It can have a probability associated to it.
- It can define a Success or a Failure, based on whether a RE outcome is a member of the event.
- It can be a member of a Random Experiment Event Set (such as an Event Space).
- It can be associated with a Random Experiment Process, e.g. the Event of Half the Trials being successes
- It can be, based on the type of Sample Space, defined as:
  - a Finite Event Set, if for a Finite Sample Space.
  - an Event Interval, if for a Continuous Sample Space.
Example(s):
- finite set events:
  - {H} ⊂ {H,T}, e.g. the |vent of a Coin turning up an "heads" in a Coin Toss Experiment.
  - {1,3,5} ⊂ {1,2,3,4,5,6}, e.g. the event of a Dice turning up an Odd Number in a Dice Experiment (with 50% probability).
- event intervals:
  - (100,300) ⊂ [0, ∞), e.g. the Event of a Lightbulb Failing between 100 and 300 Hours in a Lifetime Experiment.
  - (⅓, ⅔) ⊂ [0,1), e.g. the Event of a Gaussian Random Experiment falling within ⅔ and ⅔.
- …
Counter-Example(s):
- "H,T,T,T,T,H", a specific Binomial Trial produced by six coin-toss experiments.
- {297.39, 9201.71, 17293.27} possibly a Random Experiment Trial associated with a Random Experiment Process of Lifetime Experiments.
See: Random Experiment Outcome Set, Random Sample, Joint Distribution.

References

http://en.wikipedia.org/wiki/Event_(probability_theory)
- In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned.^[1] A single outcome may be an element of many different events,^[2] and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes.^[3] An event defines a complementary event, namely the complementary set (the event not occurring), and together these define a Bernoulli trial: did the event occur or not?
  Typically, when the sample space is finite, any subset of the sample space is an event (i.e. all elements of the power set of the sample space are defined as events). However, this approach does not work well in cases where the sample space is uncountably infinite, most notably when the outcome is a real number. So, when defining a probability space it is possible, and often necessary, to exclude certain subsets of the sample space from being events (see Events in probability spaces, below).

http://en.wikipedia.org/wiki/Event_%28probability_theory%29#A_note_on_notation
- Even though events are subsets of some sample space Ω, they are often written as propositional formulas involving random variables. For example, if X is a real-valued random variable defined on the sample space Ω, the event :[math]\displaystyle{ \{\omega\in\Omega \mid u \lt X(\omega) \leq v\}\, }[/math] can be written more conveniently as, simply, :[math]\displaystyle{ u \lt X \leq v\,. }[/math] This is especially common in formulas for a probability, such as: [math]\displaystyle{ P(u \lt X \leq v) = F(v)-F(u)\,. }[/math] The set u < X ≤ v is an example of an inverse image under the mapping X because [math]\displaystyle{ \omega \in X^{-1}((u, v]) }[/math] if and only if [math]\displaystyle{ u \lt X(\omega) \leq v }[/math].

http://www.teacherlink.org/content/math/interactive/probability/glossary/glossary.html
- Event: "A subset of a sample space" (Brown, 1997).
- Trial: A single repetition of an experiment.
- Compound Event: An event that consists of two, or more, simple events; for example: A or B; A and B and C.
- Bernoulli Trial: Another name for a trial in a binomial experiment.

(Qian) => Gang Qian. (2008). Basic Probability Theory." Lecture Notes: AME 598 Sensor Fusion, Arizona State University, Fall 2008.
- A event is a subset of the sample space S, a set of samples.
- Two special events:
  - Certain event: S
  - Impossible or null event: 

(Hogg & Ledolter, 1987) ⇒ Robert V. Hogg and Johannes Ledolter. (1987). “Engineering Statistics. Macmillan Publishing Company.
- Random experiments have outcomes that cannot be determined with certainty before the experiments are performed... The collection of all possible outcomes, namely [math]\displaystyle{ S }[/math] = {H,T}, is called the sample space. Suppose that we are interested in a subset [math]\displaystyle{ A }[/math] of our sample space; for example, in our case, let A={H} represent heads. Repeat this random experiment a number of times, say [math]\displaystyle{ n }[/math], and count the number of times, say [math]\displaystyle{ f }[/math], that the experiment ended in A. Here [math]\displaystyle{ f }[/math] is called the frequency of the event A and the ratio f/n is called the relative frequency of the 'event A in the [math]\displaystyle{ n }[/math] trials of the experiment.

(Larsen & Marx, 1986) ⇒ Richard J. Larsen, and Morris L. Marx. (1986). “An Introduction to Mathematical Statistics and Its Applications, 2nd edition." Prentice Hall
- By an experiment we will mean any procedure that (1) can be repeated, theoretically, an infinite number of times; and (2) has a well-defined set of possible outcomes. Thus, rolling a pair of dice qualifies as an experiment; so does measuring a hypertensive's blood pressure or doing a stereographic analysis to determine the carbon content of moon rocks. Each of the potential eventualities of an experiment is referred to as a sample outcome, [math]\displaystyle{ s }[/math], and their totality is called the sample space, S. To signify the member of [math]\displaystyle{ s }[/math] in [math]\displaystyle{ S }[/math], we write [math]\displaystyle{ s }[/math] In S. Any designated collection of sample outcomes, including individual outcomes, the entire sample space, and the null set, constitutes an event. The latter is said to occur if the outcome of the experiment is one of the members of that event.