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A common example of a first-hitting-time model is a ruin problem, such as Gambler's ruin. In this example, an entity (often described as a gambler or an insurance company) has an amount of money which varies randomly with time, possibly with some drift. The model considers the event that the amount of money reaches 0, representing bankruptcy.
In other problems, the objective is generating draws from a sequence of probability distributions satisfying a nonlinear evolution equation. These flows of probability distributions can always be interpreted as the distributions of the random states of a Markov process whose transition probabilities depend on the distributions of the current ...
The secretary problem demonstrates a scenario involving optimal stopping theory [1] [2] that is studied extensively in the fields of applied probability, statistics, and decision theory. It is also known as the marriage problem , the sultan's dowry problem , the fussy suitor problem , the googol game , and the best choice problem .
An important example of the use of Palm probabilities is Feller's paradox, often associated with the analysis of an M/G/1 queue. This states that the (time-)average time between the previous and next points in a point process is greater than the expected interval between points. The latter is the Palm expectation of the former, conditioning on ...
The and coefficients may be entered into the logistic regression equation to estimate the probability of passing the exam. For example, for a student who studies 2 hours, entering the value = into the equation gives the estimated probability of passing the exam of 0.25:
In probability theory and statistics, the Weibull distribution / ˈ w aɪ b ʊ l / is a continuous probability distribution. It models a broad range of random variables, largely in the nature of a time to failure or time between events. Examples are maximum one-day rainfalls and the time a user spends on a web page.
Another special case of the master equation is the Fokker–Planck equation which describes the time evolution of a continuous probability distribution. [3] Complicated master equations which resist analytic treatment can be cast into this form (under various approximations), by using approximation techniques such as the system size expansion.
Uncertainty quantification (UQ) is the science of quantitative characterization and estimation of uncertainties in both computational and real world applications. It tries to determine how likely certain outcomes are if some aspects of the system are not exactly known.
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