Random Experiment Outcome Member

A Random Experiment Outcome Member is a member of a Random Experiment Sample Space.

• AKA: Random Experiment Possible Outcome, Random Experiment Outcome, Single Random Event, Elementary Random Experiment Event, Elementary Event, Elementary Random Event, Trial, Random Trial, Single Random Experiment.
• Context:
  • It can be associated with a Random Experiment Actual Outcome (after a Random Experiment Trial Event).
  • It can be:
    • a Discrete Random Experiment Outcome, e.g. (Ace,Diamond) for a Card Draw Experiment.
    • a Continuous Random Experiment Outcome, e.g. 729.2792 for a Life Time Experiment.
  • It can be:
    • an Elementary Random Experiment Outcome, e.g. $5$, from a Dice Roll Experiment.
    • a Compound Random Experiment Outcome, e.g. $7$, from (the Sum of) Two Dice Roll Experiments.
  • It can be in a Mutually Independent Relation with another Outcome (from the same Sample Space).
  • It can be in an Independent Outcome Relation with another Outcome.
  • It can be associated to a Random Variable Member.
• Example(s):
  • Discrete Random Experiment Outcomes.
    • [math]\displaystyle{ T }[/math], from a Coin Toss Experiment.
    • $5$, from a Dice Roll Experiment.
    • (3,D), from a Card Draw Experiment.
    • $7$, from a Sum of Two Dice Rolls.
    • Success Outcome, from a Binomial Experiment.
  • Continuous Random Experiment Outcomes.
    • 7.12638, from a Lifetime Experiment.
  • …
• Counter-Example(s):
  • a Sample Space, e.g. {H,T}
  • an Impossible Event, e.g. {}.
• See: Process Possible Outcome, Random Variable Outcome, Probability Function.

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References

2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Event_(probability_theory)
  • In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. 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 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.

2008

• (Qian) => Gang Qian. (2008). Basic Probability Theory." Lecture Notes: AME 598 Sensor Fusion, Arizona State University, Fall 2008.
  • A random experiment is an experiment in which the outcome varies in a unpredictable fashion when the experiment is repeated under the same condition.
  • A random experiment is specified by stating an experimental procedure and a set of one or more measurements or observations.
  • Examples:
    • E1:Toss a coin three times and note the sides facing up (heads or tail)
    • E2:Pick a number at random between zero and one.
    • E3: Poses done by a rookie dancer
  • A sample point (o) or an outcome of a random experiment is defined as a result that cannot be decomposed into other results.

2007

• http://alea.ine.pt/english/html/glossar/html/glossar.html
  • Elementary Event: An event that corresponds to a single possible result of a random experiment.
  • Event: It is an element of the possible results of a random experiment, or, in other words, it is a subset of the sample space S.
  • Random experiment An experiment with the following characteristics: it can be repeatedly performed, in the same circumstances or in an independent manner, any time it is repeated; - the possible results are known; there is insufficient knowledge to know which result will be obtained from amongst the possible results when the experiment is performed or phenomenon observed.

2005

• Glossary of Probability Terms. http://www.teacherlink.org/content/math/interactive/probability/glossary/glossary.html
  • Experiment: An action that has various outcomes that occur unpredictably and can be repeated indefinitely under the same conditions.
  • Outcome: A result of an experiment.
  • Random Event: An event that cannot be predicted with certainty and that is chosen without any preference over other events.
  • Elementary Event: "An event that contains a single outcome" (Dolciani, Sorgenfrey, Graham, & Myers, 1988). Also called a Simple Event.
  • Compound Event: An event that consists of two, or more, simple events; for example: A or B; A and B and C.
  • Event: "A subset of a sample space" (Brown, 1997).

1987

• (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 [math]\displaystyle{ A }[/math] in the [math]\displaystyle{ n }[/math] trials of the experiment.

1986

• (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.
  • Throughout most of Chapter 3, probability functions were defined in terms of the 'elementary outcomes making up an experiment's sample space. Thus, if two fair dice were tossed, a [math]\displaystyle{ P }[/math] value was assigned to each of the 36 possible pairs of upturned faces: … 1/36 … We have already seen, though, that in certain situations some attribute of an outcome may hold more interest for the experimenter than the outcome itself. A craps player, for example, may be concerned only that he throws a 7... In this chapter we investigate the consequences of redefining an experiment's sample space. … The original sample space contains 36 outcomes, all equally likely. The revised sample space contains 11 outcomes, but the latter are not equally likely.

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