Abstract Random Experiment

An Abstract Random Experiment is a stable repeatable observable stochastic process that produces outcomes from a predefined sample space.

• AKA: Random Trial Process.
• Context:
  • It can be defined by a 2-Tuple (RET, n):
    • RET: is the Random Experiment Trial (defined by: Sample Size, Event Space, Probability Function).
    • [math]\displaystyle{ n }[/math] is a Non-Negative Integer of the number trials.
  • It can be associated with a Random Experiment Process (with [math]\displaystyle{ n }[/math] Random Experiment Trials).
  • It can range, based on the Process Outcome Set, from being a Discrete Random Experiment (for countable sets) to being a Continuous Random Experiment (for uncountable sets)
  • It can range, based on the Process Outcome Set, from being an Elementary Random Experiment to being a Complex Random Experiment.
  • It can be:
    • in a Mutually Independent Relation with another Random Experiment, and be associated with an Independent Random Experiment Process.
    • in a Conditionally Dependent Relation, with another Random Experiment, and be associated with a Conditionally Independent Random Experiment Process.
    • in a Dependent Relation with another Random Experiment, and be associated with a Dependent Random Experiment Process.
  • It can be represented by a Random Variable.
  • ? It can be Modelled with a Statistical Model (that identifies Dependent Variables and Independent Variables).
• Example(s):
  • any Elementary Discrete Random Experiment
    • A two Coin Toss Experiment: E.g. Toss a coin (with symbols {H,T}) two consecutive times, and note the symbol on coin's exposed side.
    • A Dice Roll Experiment: E.g. Toss a dice (with symbols: {1,2,3,4,5,6}) three consecutive times, and note whether the three numbers add up to 9 or not.
    • A Card Draw Experiment: E.g. Select five cards (with 52 symbol pairs: {(ace,heart),...,(king,diamond)}), and note whether the set is a "flush", a "straight", a "straight flush", is "four of a kind", or other.
  • any Composite Discrete Random Experiment, such as the Sum of Two Dice Rolls.
  • any Elementary Continuous Random Experiment, such as a lightbulb lifetime experiment.
  • Complex Continuous Random Experiment, such as from a complex lifetime experiment.
  • …
• Counter-Example(s):
  • any Deterministic Process.
  • a Non-Stable Stochastic Process (where the Probability Function can change from Outcome to Outcome).
  • a Non-Repeatable Stochastic Process.
  • The set of Outcomes of a Two Dice Roll Experiment at a Craps table over the next hour (because the number of trials is not fixed).
  • an unbounded Sequence of Random Experiments.
    • the decay of every uranium atom.
  • a string of different Random Experiments.
    • a series of two Dice Rolls or two Card Draws decided by whether a Coin Flip=H.
  • any Randomized Controlled Experiment.
• See: Process, Multinomial Process.

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References

2011

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Experiment_%28probability_theory%29
  • QUOTE: An experiment is any procedure that can be infinitely repeated and has a well defined set of outcomes. Examples include tossing a coin or rolling a die. In probability theory, an experiment refers to random experiment. An experiment is said to be random experiment if it has more than one possible outcomes. If an experiment has only one possible outcome then it is known as deterministic experiment. An experiment is composed of one or more trials. A trial with two mutually exclusive outcomes is known as Bernoulli trial.

• (Forbes et al., 2011) ⇒ Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). Statistical distributions. John Wiley & Sons.
  • A probabilistic experiment is some occurrence such as the tossing of coins, rolling dice or observation of rainfall on a particular day where a complex natural background leads to a chance outcome.

2009

• http://alea.ine.pt/english/html/glossar/html/glossar.html
  • QUOTE: 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.

2008

• (Qian, 2008) ⇒ Gang Qian. (2008). Basic Probability Theory.” Lecture Notes: AME 598 Sensor Fusion, Arizona State University, Fall 2008.
  • QUOTE: A random experiment is an experiment in which the outcome varies in a unpredictable fashion when the experiment is repeated under the same condition.
  • QUOTE: A random experiment is specified by stating an experimental procedure and a set of one or more measurements or observations.
  • QUOTE: 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

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.

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