Random Poisson distribution

Nature In monitoring a moving threadhne, one criterion of quality would be the frequency of broken filaments. These can be identified as they occur through the threadhne by a broken-filament detector mounted adjacent to the threadhne. In this context, the random occurrences of broken filaments can be modeled by the Poisson distribution. This is called a Poisson process and corresponds to a probabilistic description of the frequency of defects or, in general, what are called arrivals at points on a continuous line or in time. Other examples include ... 

Here, f(x) is tlie probability of x occurrences of an event tliat occurs on the average p times per unit of space or time. Both tlie mean and tlie variance of a random variable X liaving a Poisson distribution are (i. 

The Poisson distribution can be used to determine probabilities for discrete random variables where the random variable is the number of times that an event occurs in a single trial (unit of lime, space, etc.). The probability function for a Poisson random variable is... 

Independent gaussian random variables are by no means the only ones whose distributions are preserved under addition. Another example is independent, Poisson distributed random variables, for which... 

On the other hand, Eq. (3-233) states that A is the sum of two statistically independent, Poisson distributed random variables Ax and Aa with parameters n(t2 — tj) and n tx — t2) respectively. Consequently,49 A must be Poisson distributed with parameter n(t2 — tx) + n(t3 — t2) = n(t3 — tx) which checks our direct calculation. The fact that the most general consistency condition of the type just considered is also met follows in a similar manner from the properties of sums of independent, Poisson distributed random variables. 

Fig. 4. A schematic two-dimensional illustration of the idea for an information theory model of hydrophobic hydration. Direct insertion of a solute of substantial size (the larger circle) will be impractical. For smaller solutes (the smaller circles) the situation is tractable a successful insertion is found, for example, in the upper panel on the right. For either the small or the large solute, statistical information can be collected that leads to reasonable but approximate models of the hydration free energy, Eq. (7). An important issue is that the solvent configurations (here, the point sets) are supplied by simulation or X-ray or neutron scattering experiments. Therefore, solvent structural assumptions can be avoided to some degree. The point set for the upper panel is obtained by pseudo-random-number generation so the correct inference would be of a Poisson distribution of points and = kTpv where v is the van der Waals volume of the solute. Quasi-random series were used for the bottom panel so those inferences should be different. See Pratt et al. (1999).

When retention ordering can be established, the theoretical peak capacity could be effectively utilized in a multidimensional separation system in a far more efficient manner. However, one is reminded that with the exception of synthetic polymers and a few other special cases of small molecules, real samples have almost random retention time distributions. It is rare when the free energy, enthalpy, and entropy of interaction are determined in LC for molecules utilized in retention mechanism studies. However, the retention energetics have been determined in GC studies by Davis et al. (2000) who found that many complex samples will exhibit Poisson distributions of retention times due to a Poisson distribution in enthalpy and a compensating distribution in entropy. 

Similar mathematical solution can be derived from a Poisson distribution of random events in 2D space. The probability that 2D separation space will be covered by peaks in ideally orthogonal separation is analogical to an example where balls are randomly thrown in 2D space divided into uniform bins. The general relationship between the number of events K (number of balls, peaks, etc.) and the number of bins occupied F (bins containing one or more balls, peaks, etc.) is described by Equation 12.3, where N is the number of available bins (peak capacity in 2DLC). 

Figure 6. Formfactor K(r) and nearest neighbour spacing distribution P(s) for (a) the pi s form the regular representation of the cyclic group Z24 (b) pi s represent the symmetric group. S, (c) a random set of pi s without symmetries. The dashed curve in (b) labeled red. Poisson corresponds to a distribution of degenerate levels being Poisson distributed otherwise.

Let us now look into two examples to get an impression of a compound demand. We will look at compound Poisson distributions. Poisson distributions describe the random number of independent events per period, for example the number of customers with nonzero demands in a certain week from a large customer base. 

At any one instant, only a very small proportion of the total number of unstable nuclei in a radioactive source undergo decay. A Poisson distribution which expresses the result of a large number of experiments in which only a small number are successful, can thus be used to describe the results obtained from measurements on a source of constant activity. In practical terms this means that random fluctuations will always occur, and that the estimated standard deviation, 5, of a measurement can be related to the total measurement by ... 

In LSC measurements precautions are required to avoid impurities which may cause scintillation quenching. Since radioactive decay is random and is described with the Poisson distribution, the standard deviation for a given count, C, is equal to C1/2. 

To avoid penetration and filament formation via static and randomly scattered pinholes, one approach is to diminish the area of the junction until statistically the presence of a pinhole defect is near vanishing, as might be calculated by a Poisson distribution. An example of this strategy is the use of a nanopore junction of 50 nm diameter, though in this case the device fabrication yields were still reported to be quite low, down to a few percent [16]. 

The Poisson distribution describes the results of experiments in which we count events that occur at random but at a definite average rate. Examples of the Poisson distribution include the number of emails we receive in a one-day period, the number of babies bom in a hospital in a two-day period, the number of decays of a radioactive isotope in a one-day period. 

It is a well recognized fact that in field ion microscopy field evaporation does not occur at a constant rate because of the atomic step structures of the tip surface. For the sole purpose of a compositional analysis of a sample, one should try to aim the probe hole at a high index plane where the step height is small and field evaporation occurs more uniformly. But even so, the number of atoms field evaporated per HV pulse or laser pulse within the area covered by the probe-hole will not be the same every time. It is reasonable to assume that the field evaporation events are nearly random even though there has been no systematic study of the nature of such field evaporation events. Let the average number of atoms field evaporated per pulse within the area covered by the probe-hole area be n. The probability that n atoms are field evaporated by a pulse is then given by the Poisson distribution... 

Suppose an inspection unit of a certain product is selected and examined from a process running with a stable nonconformity rate c per inspection unit and X nonconformities are found.Then Vis a random variable following a Poisson distribution with parameter c. If the true nonconformity level c is known, then the parameters of the c chart are... 

Q = [Q]/[M], If Q is randomly distributed among the micelles, then the probability that a particular micelle has n molecules of Q is given by the Poisson distribution 23... 

A stochastic theory provides a simple model for chromatography.11 The term stochastic implies the presence of a random variable. The model supposes that, as a molecule travels through a column, it spends an average time Tm in the mobile phase between adsorption events. The time between desorption and the next adsorption is random, but the average time is Tm. The average time spent adsorbed to the stationary phase between one adsorption and one desorption is rs. While the molecule is adsorbed on the stationary phase, it does not move. When the molecule is in the mobile phase, it moves with the speed ux of the mobile phase. The probability that an adsorption or desorption occurs in a given time follows the Poisson distribution, which was described briefly in Problem 19-21. 

A Poisson distribution is valid when (a) all possible outcomes are random and independent of one another, (b) the maximum possible value of n is a large number, and (c) the average value of n is a small fraction of the maximum possible value. 

Exercise. Suppose there are N primary individuals, where N is random with a Poisson distribution. Each primary produces M secondaries, where M is again Poissonian. Find an expression for the distribution of the total number of individuals. Exercise. Consider a superposition of Poisson distributions ... 

It is understood that P1(1 = 0 for n2 < n. Thus each sample function y(t) is a succession of steps of unit height and at random moments. It is uniquely determined by the time points at which the steps take place. These time points constitute a random set of dots on the time axis. Their number between any two times tl912 is distributed according to the Poisson distribution (2.6). Hence Y(t) is called Poisson process and describes the same situation as (II.2.6). 

Remark. The white noise limit is not sufficiently defined by just saying rc 0. We have to construct a sequence of processes which in this limit reduce to Gaussian white noise. For that purpose take a long time interval (0, T) and a Poisson distribution of time points Ta in it with density v. To each Ta attach a random number ca they are independent and identically distributed, with zero mean. Consider the process... 

Fig. 8. Random crosslinking of monodisperse chains and aPoisson distribution of primary chains ve jy 1 Poisson distribution and monodisperse chains, Pn- oo 2 monodisperse chains jP = 10.Pci/P 3 monodisperse chain and Poisson distribution, Pn-> oo 4 Poisson distribution, Pn = 10 5 monodisperse chains P = 10. The dotted lines show the gel point and full conversion for Pn = 10 [Dobson and Gordon (4 )]...

Solution. Poisson statistics apply when events are random and mutually independent, which is assumed to be the case both in time and along the wire. The probability p(n, A) that n events occur in an area" (length x time) A with event rate J is given by the Poisson distribution... 

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