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In probability, a singular distribution is a probability distribution concentrated on a set of Lebesgue measure zero, where the probability of each point in that set is zero.
These distributions are sometimes called singular continuous distributions, since their cumulative distribution functions are singular and continuous.
Such distributions are not absolutely continuous with respect to Lebesgue measure.
A singular distribution is not a discrete probability distribution because each discrete point has a zero probability. On the other hand, neither does it have a probability density function, since the Lebesgue integral of any such function would be zero.
In general, distributions can be described as a discrete distribution (with a probability mass function), an absolutely continuous distribution (with a probability density), a singular distribution (with neither), or can be decomposed into a mixture of these.
An example is the Cantor distribution; its cumulative distribution function is a devil's staircase. Less curious examples appear in higher dimensions. For example, the upper and lower Fréchet–Hoeffding bounds are singular distributions in two dimensions.
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In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable , or just distribution function of , evaluated at , is the probability that will take a value less than or equal to .
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion. Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem. Although having a long history, an axiomatization was done during the beginning and mid of the 20th century. During that time, the connection between probability and statistic was established, thereby also connecting to the field of measure theory. In its modern form, it developed mostly in parallel to more prominent physical theories and is often overshadowed by them, and therefore sometimes leading to inconsistent theories or wrong beliefs.
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