In
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
and
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, a realization, observation, or observed value, of a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
is the value that is actually
observed (what actually happened). The random variable itself is the process dictating how the observation comes about. Statistical quantities computed from realizations without deploying a statistical model are often called "
empirical
Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...
", as in
empirical distribution function
In statistics, an empirical distribution function (commonly also called an empirical Cumulative Distribution Function, eCDF) is the distribution function associated with the empirical measure of a sample. This cumulative distribution function ...
or
empirical probability The empirical probability, relative frequency, or experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials, not in a theoretical sample space but in an actual experi ...
.
Conventionally, to avoid confusion, upper case letters denote random variables; the corresponding lower case letters denote their realizations.
Formal definition
In more formal
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, a random variable is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
''X'' defined from a
sample space
In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...
Ω to a measurable space called the state space.
If an element in Ω is mapped to an element in state space by ''X'', then that element in state space is a realization. Elements of the sample space can be thought of as all the different possibilities that ''could'' happen; while a realization (an element of the state space) can be thought of as the value ''X'' attains when one of the possibilities ''did'' happen.
Probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
is a
mapping that assigns numbers between zero and one to certain
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of the sample space, namely the measurable subsets, known here as
events
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of ev ...
. Subsets of the sample space that contain only one element are called
elementary event
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 an ...
s. The value of the random variable (that is, the function) ''X'' at a point ω ∈ Ω,
:
is called a realization of ''X''.
See also
*
Errors and residuals
In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "true value" (not necessarily observable). The err ...
*
Outcome (probability)
In probability theory, an outcome is a possible result of an experiment or trial. Each possible outcome of a particular experiment is unique, and different outcomes are mutually exclusive (only one outcome will occur on each trial of the experimen ...
*
Random variate
In probability and statistics, a random variate or simply variate is a particular outcome of a ''random variable'': the random variates which are other outcomes of the same random variable might have different values ( random numbers).
A random ...
*
Raw data
Raw data, also known as primary data, are ''data'' (e.g., numbers, instrument readings, figures, etc.) collected from a source. In the context of examinations, the raw data might be described as a raw score (after test scores).
If a scientist ...
Notes
References
{{Reflist
Statistical data types