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  2. Random sequence - Wikipedia

    en.wikipedia.org/wiki/Random_sequence

    Traditional probability theory does not state if a specific sequence is random, but generally proceeds to discuss the properties of random variables and stochastic sequences assuming some definition of randomness. The Bourbaki school considered the statement "let us consider a random sequence" an abuse of language. [3]

  3. Randomness - Wikipedia

    en.wikipedia.org/wiki/Randomness

    Individual random events are, by definition, unpredictable, but if there is a known probability distribution, the frequency of different outcomes over repeated events (or "trials") is predictable. [note 1] For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often ...

  4. Convergence of random variables - Wikipedia

    en.wikipedia.org/wiki/Convergence_of_random...

    The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than others. For example, convergence in distribution tells us about the limit distribution of a sequence of random variables. This is a weaker notion than convergence in probability, which tells us about the ...

  5. Independent and identically distributed random variables

    en.wikipedia.org/wiki/Independent_and...

    A random sample can be thought of as a set of objects that are chosen randomly. More formally, it is "a sequence of independent, identically distributed (IID) random data points." In other words, the terms random sample and IID are synonymous. In statistics, "random sample" is the typical terminology, but in probability, it is more common to ...

  6. Stochastic process - Wikipedia

    en.wikipedia.org/wiki/Stochastic_process

    A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one.

  7. Martingale (probability theory) - Wikipedia

    en.wikipedia.org/wiki/Martingale_(probability...

    In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values. Stopped Brownian motion is an example of a martingale. It can model an even coin-toss ...

  8. Statistical randomness - Wikipedia

    en.wikipedia.org/wiki/Statistical_randomness

    In a "truly" random sequence of numbers of sufficient length, for example, it is probable there would be long sequences of nothing but repeating numbers, though on the whole the sequence might be random. Local randomness refers to the idea that there can be minimum sequence lengths in which random distributions are approximated. Long stretches ...

  9. Event (probability theory) - Wikipedia

    en.wikipedia.org/wiki/Event_(probability_theory)

    An event, however, is any subset of the sample space, including any singleton set (an elementary event), the empty set (an impossible event, with probability zero) and the sample space itself (a certain event, with probability one). Other events are proper subsets of the sample space that contain multiple elements. So, for example, potential ...