Sample Space

A Sample Space is a multiset that represents all possible outcomes of a random experiment.

• AKA: S, Ω, U, Random Experiment Event Space, Sample Description Space, Possibility Space.
• Context:
  • It can be associated with a Probability Space, Probability Measure and Random Experiment Event while Statistical Sample can be associated with a Statistical Population, Sampling Distribution and Statistical Hypothesis Testing.
  • It can range from being a Discrete Sample Space (e.g. categorical sample space) to being a Continuous Sample Space (e.g a numeric sample space).
  • It can range from being an Ordered Sample Space to being an Unordered Sample Space.
  • It can be a Superset of a Random Experiment Event.
• Example(s):
  • Ω = {Heads, Tails}, the Discrete Sample Space associated with a Coin Toss Experiment.
  • Ω = (One, Two, Three, Four, Five, Six), the Discrete Sample Space associated with the Random Experiment of throwing a die once has a sample space with six outcomes.
  • Ω = {(Ace,Diamonds), ..., (King,Hearts)} , associated with a one-card draw.
  • …
• Counter-Example(s):
  • a Multiset, such as {Heads, Tails, Heads}.
  • Statistical Sample.
  • Statistical Population.
• See: Probability Space, Probability Theory, Probability Measure, Random Experiment Event, Sigma-Algebra.

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References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Sample_space
  • In probability theory, the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible outcomes are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U (for “universal set").

For example, if the experiment is tossing a coin, the sample space is typically the set {head, tail}. For tossing two coins, the corresponding sample space would be {(head,head), (head,tail), (tail,head), (tail,tail)}. For tossing a single six-sided die, the typical sample space is {1, 2, 3, 4, 5, 6} (in which the result of interest is the number of pips facing up).

A well-defined sample space is one of three basic elements in a probabilistic model (a probability space); the other two are a well-defined set of possible events (a sigma-algebra) and a probability assigned to each event (a probability measure function).

• (OECD, 2016) ⇒ Glossary of Statistical Terms http://stats.oecd.org/glossary/detail.asp?ID=3855
  • The set of sample points corresponding to all possible samples. The permissible domain of variation of a sample point. Sometimes referred to as sample description space or event space.

• (probabilityformula.org, 2016) ⇒ http://www.probabilityformula.org/sample-space.html
  • The sample space plays a very important role in finding the probability of an event associated with a random experiment. Sample space is a term associated with probability theory.

In an experiment or any random trial, when we make a set of all the results or outcomes that are possible in that experiment or trial, that set is said to be a sample space of that particular experiment. For example, when we flip a coin there are only two possibilities; either to get a head or a tail. If we represent head by ‘H’ and tail be ‘T’ then the sample space of the experiment of flipping a coin, say S = {H, T}.

To find the probability of any event associated with a random experiment we divide the number of outcomes of that event with the total number of possible outcomes of the whole experiment that is the sample space.(...) In general sample space is denoted by three common symbols: U, S, Ω.

Commonly we used, ‘U’ to represents universal set. An experiment can have more than one sample space depending on the condition been set.

• http://en.wiktionary.org/wiki/sample_space
  • The set of all possible outcomes of a game, experiment or other situation
• http://en.wikibooks.org/wiki/SA_NCS_Mathematics:Glossary
  • the collection of all the possible outcomes in a statistical experiment; may be discrete (consisting of categorised or counted values) or continuous (when a measurement is made on a scale that is continuous, eg mass, temperature, height).
• http://www.introductorystatistics.com/escout/main/Glossary.htm
  • sample space The set of all possible outcomes of a probability experiment. For example, the probability experiment of throwing a die once has the sample space S = (1,2,3,4,5, 6).

2011

• (Forbes et al., 2011) ⇒ Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). Statistical distributions. John Wiley & Sons.
  • The set of possible out comes of a probabilistic experiment is called the sample, event, or possibility space. For example, if two coins are tossed, the sample space is the set of possible results HH, HT, TH, and TT, where H indicates a head and T a tail.

2008

• (Qian, 2008) => Gang Qian. (2008). Basic Probability Theory." Lecture Notes: AME 598 Sensor Fusion, Arizona State University, Fall 2008.
  • QUOTE: Sample Space (S): defined as the set of all possible outcomes from a random experiment.
  • QUOTE: Countable or discrete sample space, one-to-one correspondence between outcomes and integers.
  • QUOTE: Uncountable or continuous sample space.
  • QUOTE: Probability Space {S,E,P}
    • S: Sample space, the space of the outcomes from a random experiment {o}
    • E: Event space, collection of subsets of the sample space, {A}
    • P: Probability measure of a event P(A), ranges [0,1], encoding how likely an event will happen.

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 s∈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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