Waiting-time probability density function

Here, we present an approach for the description of such anomalous transport processes that is based on the continuous-time random walk theory for a power-law waiting time distribution w(t) but which can be used to find the probability density function of the random walker in the presence of an external force field, or in phase space. This framework is fractional dynamics, and we show how the traditional kinetic equations can be generalized and solved within this approach. 

Particles, such as molecules, atoms, or ions, and individuals, such as cells or animals, move in space driven by various forces or cues. In particular, particles or individuals can move randomly, undergo velocity jump processes or spatial jump processes [333], The steps of the random walk can be independent or correlated, unbiased or biased. The probability density function (PDF) for the jump length can decay rapidly or exhibit a heavy tail. Similarly, the PDF for the waiting time between successive jumps can decay rapidly or exhibit a heavy tail. We will discuss these various possibilities in detail in Chap. 3. Below we provide an introduction to three transport processes standard diffusion, tfansport with inertia, and anomalous diffusion. 

Quantification of polymer entropies is done through the statistics of random walk (Flory, 1953). The model is based on a drunk trying to walk in one dimension Because of his state, the next step the drunk takes could be to the right or to the left with equal probability, but his stride rranains of idmtical length and at every step he waits for the same length of time. The key probability density function is that of end-to-end displacement x, that is, the distance betweai the beginning and the end, which for a linear polymer in one dimraision is Ganssian ... 

Disorder was introduced into this system by postulating a distribution of waiting times. A complementary extension of the theory may be made by considering a distribution of jump distances. It may be shown that, as a consequence of the central limit theorem, provided the single-step probability density function has a finite second moment, Gaussian diffusion is guaranteed. If this condition is not satisfied, however, then Eq. (105) must be replaced by... 

Suppose a random set of dots representing a sequence of events is given. The following question may be asked. If I start observing at some time t0, how long do I have to wait for the next event to occur Of course, the time 6 from t0 to the next event is a random variable with values in (0, oo) and the quantity of interest is its probability density, w(6 t0) (which depends parametrically on t0 unless the random set of events is stationary). This question is of particular interest in queuing problems. The function w(6 t0) has also been measured electronically for the arrivals of photons produced by luminescence. 

Exponential Distribution A third probability distribution arising often in computational biology is the exponential distribution. This distribution can be used to model lifetimes, analogous to the use of the geometric distribution in the discrete case as such it is an example of a continuous waiting time distribution. A r.v. X with this distribution [denoted by X exp(A)] has range [0, - - >] and density function... 

In practical applications, the maximum likelihood estimate (MLE) is by far the most used technique to evaluate the parameter s models. The MLE is based on the use of the likelihood function L for the estimation of parameters. The L function represents the joint probability density of the analyzed variable, for example, the waiting times or the inter-arrival times. Given N data points and a model whose parameters are... 

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