Realization (probability)

In probability and statistics, a realization or observation (also called observed value) of a random variable or random element is the value that is actually observed or measured. For example, if the random variable is human height, a given realization could be "2 meters”. 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", as in empirical distribution function or empirical probability. Conventionally, to avoid confusion, upper case letters denote random variables and the corresponding lower case letters denote their realizations.

In probability and statistics, a realization or observation (also called observed value) of a random variable or random element is the value that is actually observed or measured. For example, if the random variable is human height, a given realization could be "2 meters”. 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", as in empirical distribution function or empirical probability. Conventionally, to avoid confusion, upper case letters denote random variables and the corresponding lower case letters denote their realizations. In probability theory, a random variable is a function X defined from a sample space \Omega to another measurable space called the state space . If an element in \Omega is mapped to an element in state space by X , then that element in state space is a realization. That is, at a point \omega\in\Omega , the value x = X(\omega) is called a realization of X . 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 is a mapping that assigns numbers between zero and one to certain subsets of the sample space, namely the measurable subsets, known here as events. Subsets of the sample space that contain only one element are called elementary events.

See also

Data point (statistics) Errors and residuals Outcome (probability) Random variate Raw data Deviation (statistics) Raw score Random number generation Non-uniform random number generation Pseudo-random number sampling

Notes

References

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