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In probability theory and statistics, the probability distribution of a mixed random variable consists of both discrete and continuous components. A mixed random variable does not have a cumulative distribution function that is discrete or everywhere-continuous. An example of a mixed type random variable is the probability of wait time in a queue.
Discrete time is often employed when empirical measurements are involved, because normally it is only possible to measure variables sequentially. For example, while economic activity actually occurs continuously, there being no moment when the economy is totally in a pause, it is only possible to measure economic activity discretely.
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic.
A discrete power-law distribution, the most famous example of which is the description of the frequency of words in the English language. The Zipf–Mandelbrot law is a discrete power law distribution which is a generalization of the Zipf distribution. Conway–Maxwell–Poisson distribution Poisson distribution Skellam distribution
For example, the sample space of a coin flip could be Ω = {"heads", "tails" }. To define probability distributions for the specific case of random variables (so the sample space can be seen as a numeric set), it is common to distinguish between discrete and absolutely continuous random variables.
Example. A singular continuous measure. The Cantor distribution has a cumulative distribution function that is continuous but not absolutely continuous , and indeed its absolutely continuous part is zero: it is singular continuous.
An extreme example: if a set X is given the discrete topology (in which every subset is open), all functions : to any topological space T are continuous. On the other hand, if X is equipped with the indiscrete topology (in which the only open subsets are the empty set and X ) and the space T set is at least T 0 , then the only continuous ...
Furthermore, it covers distributions that are neither discrete nor continuous nor mixtures of the two. An example of such distributions could be a mix of discrete and continuous distributions—for example, a random variable that is 0 with probability 1/2, and takes a random value from a normal distribution with probability 1/2.