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  2. Independence (probability theory) - Wikipedia

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

    Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.

  3. Event (probability theory) - Wikipedia

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

    In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. [1] A single outcome may be an element of many different events, [2] and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. [3]

  4. Entropy (information theory) - Wikipedia

    en.wikipedia.org/wiki/Entropy_(information_theory)

    The entropy of the unknown result of the next toss of the coin is maximized if the coin is fair (that is, if heads and tails both have equal probability 1/2). This is the situation of maximum uncertainty as it is most difficult to predict the outcome of the next toss; the result of each toss of the coin delivers one full bit of information.

  5. Contingency table - Wikipedia

    en.wikipedia.org/wiki/Contingency_table

    In other words, the two variables are not independent. If there is no contingency, it is said that the two variables are independent. The example above is the simplest kind of contingency table, a table in which each variable has only two levels; this is called a 2 × 2 contingency table. In principle, any number of rows and columns may be used.

  6. Pairwise independence - Wikipedia

    en.wikipedia.org/wiki/Pairwise_independence

    Pairwise independence does not imply mutual independence, as shown by the following example attributed to S. Bernstein. [3]Suppose X and Y are two independent tosses of a fair coin, where we designate 1 for heads and 0 for tails.

  7. Hidden Markov model - Wikipedia

    en.wikipedia.org/wiki/Hidden_Markov_model

    Figure 1. Probabilistic parameters of a hidden Markov model (example) X — states y — possible observations a — state transition probabilities b — output probabilities. In its discrete form, a hidden Markov process can be visualized as a generalization of the urn problem with replacement (where each item from the urn is returned to the original urn before the next step). [7]

  8. Likelihood function - Wikipedia

    en.wikipedia.org/wiki/Likelihood_function

    This follows from the definition of independence in probability: the probabilities of two independent events happening, given a model, is the product of the probabilities. This is particularly important when the events are from independent and identically distributed random variables , such as independent observations or sampling with replacement .

  9. Large deviations theory - Wikipedia

    en.wikipedia.org/wiki/Large_deviations_theory

    In the above example of coin-tossing we explicitly assumed that each toss is an independent trial, and the probability of getting head or tail is always the same. Let X , X 1 , X 2 , … {\displaystyle X,X_{1},X_{2},\ldots } be independent and identically distributed (i.i.d.) random variables whose common distribution satisfies a certain growth ...