Inprobability andstatistics, arealization,observation, orobserved value, of arandom variable is the value that is actuallyobserved (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", as inempirical distribution function orempirical probability.

Conventionally, to avoid confusion, upper case letters denote random variables; the corresponding lower case letters denote their realizations.^[1]

In more formalprobability theory, a random variable is afunctionX defined from asample space Ω to ameasurable space called thestate space.^[2][a] If an element in Ω is mapped to an element in state space byX, then that element in state space is a realization. Elements of the sample space can be thought of as all the different possibilities thatcould happen; while a realization (an element of the state space) can be thought of as the valueX attains when one of the possibilitiesdid happen.Probability is amapping that assigns numbers between zero and one to certainsubsets of the sample space, namely the measurable subsets, known here asevents. Subsets of the sample space that contain only one element are calledelementary events. The value of the random variable (that is, the function)X at a point ω ∈ Ω,

${\displaystyle x=X(\omega )}$

is called arealization ofX.^[3]

Inprobability andstatistics, a random variate or simply variate is a particular outcome orrealization of arandom variable; the random variates which are other outcomes of the same random variable might have different values (random numbers).^[4] Random variates are used whensimulating processes driven by random influences (stochastic processes).

Devroye defines a random variate generation algorithm (forreal numbers) as follows:^[5]

Assume that

1. Computers can manipulate real numbers.
2. Computers have access to a source of random variates that areuniformly distributed on theclosed interval [0,1].

Then a random variate generation algorithm is any program that haltsalmost surely and exits with a real numberx. Thisx is called arandom variate.

Note that both assumptions are violated in most real computers. Computers necessarily lack the ability to manipulate real numbers, typically usingfloating point representations instead. Most computers lack a source of true randomness (like certainhardware random number generators), and instead usepseudorandom number sequences.

The distinction betweenrandom variable andrandom variate is subtle and is not always made in the literature. It is useful when one wants to distinguish between a random variable itself with an associatedprobability distribution on the one hand, and random draws from that probability distribution on the other, in particular when those draws are ultimately derived byfloating-point arithmetic from a pseudo-random sequence.

• 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

• Statistical concepts

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