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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.
The Yule model makes use of two related Yule processes, where a Yule process is defined as a continuous time birth process which starts with one or more individuals. Yule proved that when time goes to infinity, the limit distribution of the number of species in a genus selected uniformly at random has a specific form and exhibits a power-law ...
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 die, a casino roulette, or the first card of a well-shuffled deck.
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.
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.
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.
Variable speedism may also be distinguished one of two ways: "discrete variable speedism" and "continuously variable speedism". Eldredge and Gould, proposing that evolution jumps between stability and relative rapidity, are described as "discrete variable speedists", and "in this respect they are genuinely radical."
Consider a continuous-time Markov process with m + 1 states, where m ≥ 1, such that the states 1,...,m are transient states and state 0 is an absorbing state. Further, let the process have an initial probability of starting in any of the m + 1 phases given by the probability vector (α 0,α) where α 0 is a scalar and α is a 1 × m vector.