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  2. Bernoulli distribution - Wikipedia

    en.wikipedia.org/wiki/Bernoulli_distribution

    In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, [1] is the discrete probability distribution of a random variable which takes the value 1 with probability and the value 0 with probability =.

  3. Bernoulli trial - Wikipedia

    en.wikipedia.org/wiki/Bernoulli_trial

    Random variables describing Bernoulli trials are often encoded using the convention that 1 = "success", 0 = "failure". Closely related to a Bernoulli trial is a binomial experiment, which consists of a fixed number n {\displaystyle n} of statistically independent Bernoulli trials, each with a probability of success p {\displaystyle p} , and ...

  4. Bernoulli process - Wikipedia

    en.wikipedia.org/wiki/Bernoulli_process

    The term Bernoulli sequence is often used informally to refer to a realization of a Bernoulli process. However, the term has an entirely different formal definition as given below. Suppose a Bernoulli process formally defined as a single random variable (see preceding section). For every infinite sequence x of coin flips, there is a sequence of ...

  5. List of probability distributions - Wikipedia

    en.wikipedia.org/wiki/List_of_probability...

    This does not look random, but it satisfies the definition of random variable. This is useful because it puts deterministic variables and random variables in the same formalism. The discrete uniform distribution, where all elements of a finite set are equally likely. This is the theoretical distribution model for a balanced coin, an unbiased ...

  6. Probability theory - Wikipedia

    en.wikipedia.org/wiki/Probability_theory

    For example, if ,,... are independent Bernoulli random variables taking values 1 with probability p and 0 with probability 1-p, then () = for all i, so that ¯ converges to p almost surely. Central limit theorem

  7. De Finetti's theorem - Wikipedia

    en.wikipedia.org/wiki/De_Finetti's_theorem

    A random variable X has a Bernoulli distribution if Pr(X = 1) = p and Pr(X = 0) = 1 − p for some p ∈ (0, 1).. De Finetti's theorem states that the probability distribution of any infinite exchangeable sequence of Bernoulli random variables is a "mixture" of the probability distributions of independent and identically distributed sequences of Bernoulli random variables.

  8. Independent and identically distributed random variables

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

    For example, a sequence of Bernoulli trials is interpreted as the Bernoulli process. One may generalize this to include continuous time Lévy processes , and many Lévy processes can be seen as limits of i.i.d. variables—for instance, the Wiener process is the limit of the Bernoulli process.

  9. Stochastic simulation - Wikipedia

    en.wikipedia.org/wiki/Stochastic_simulation

    A binomial distributed random variable Y with parameters n and p is obtained as the sum of n independent and identically Bernoulli-distributed random variables X 1, X 2, ..., X n [4] Example: A coin is tossed three times. Find the probability of getting exactly two heads. This problem can be solved by looking at the sample space.