Poisson distribution/statistics

At high concentrations of pyrene (112), where there is a substantial probability of locating more than one pyrene molecule in the same micelle, excimer emission is the dominant photophysical process to follow excitation (Figure 10). The system may be treated quantitatively by Poisson distribution statistics and applied successfully to Si decay in surfactant (112). A similar treatment of naphthalene fluorescence in micelles has been presented. ... 

When the rate measurement is statistically distributed about the mean, the distribution of events can be described by the Poisson distribution, Prrf O, given by... 

The applicability of the Poisson distribution to counting statistics can be proved directly that is, without reference to binomial theorem or Gaussian distribution. See J. L. Doob, Stochastic Processes, page 398. The standard deviation of a Poisson distribution is always the square root of its mean. 

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. 

Eq. (32) may be deduced from kinetic arguments.This is unnecessary, however, since the conventional derivation of the Poisson distribution proceeds from equivalent statistical conditions. 

The denominator in (3.1) can be simplified because the statistical uncertainty of the baseline, hN o, is negligible in practice when the spectra are simulated with numerical line fit routines. The stochastic emission of y-rays by the source leads to a Poisson distribution of counts with the width AA = and since is small, the denominator of (3.1) can be written as ... 

If the errors were greater than 10% and dominated by counting statistics, a Poisson distribution would yield slightly more accurate results. 

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).

The use of the Poisson distribution for this purpose predates the statistical overlap theory of Davis and Giddings (1983), which also utilized this approach, by 9 years. Connors work seems to be largely forgotten because it is based on 2DTLC that doesn t have the resolving power (i.e., efficiency or the number of theoretical plates) needed for complex bioseparations. However, Martin et al. (1986) offered a more modem and rigorous theoretical approach to this problem that was further clarified recently (Davis and Blumberg, 2005) with computer simulation techniques. Clearly, the concept and mathematical approach used by Connors were established ahead of its time. 

The statistical growth of a constant number of chains in a living polymerization leads to the narrow Poisson distribution of the mole fraction as a function of the degree of polymerization N ... 

Fig. 7.7. Effects of Poisson photon noise on calculated SE and FRET values. (A) Statistical distribution of number of incoming photons for the mean fluorescence intensities of 5,10, 20, 50, and 100 photons/pixel, respectively. For n = 100 (rightmost curve), the SD is 10 thus the relative coefficient of variation (RCV this is SD/mean) is 10 %. In this case, 95% of observations are between 80 and 120. For example, n — 10 the RCY has increased to 33%. (B) To visualize the spread in s.e. caused by the Poisson distribution of pixel intensities that averaged 100 photons for each A, D, and S (right-most curve), s.e. was calculated repeatedly using a Monte Carlo simulation approach. Realistic correction factors were used (a = 0.0023,/ = 0.59, y = 0.15, <5 = 0.0015) that determine 25% FRET efficiency. Note that spread in s.e. based on a population of pixels with RCY = 10 % amounts to RCV = 60 % for these particular settings Other curves for photon counts decreasing as in (A), the uncertainty further grows and an increasing fraction of calculated s.e. values are actually below zero. (C) Spread in Ed values for photon counts as in (A). Note that whereas the value of the mean remains the same, the spread (RCV) increases to several hundred percent. (D) Spread depends not only on photon counts but also on values of the correction...

In single-photon counting experiments, the statistics obey a Poisson distribution and the expected deviation [Pg.182]

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 has important applications in isotope geochemistry where counting statistics is needed. One example in... 

Exercise. When Nx and N2 are two statistically independent variables, each with a Poisson distribution, their sum + N2 is again Poissonian. 

Exercise. Find the cumulants of U for the case that the xa are statistically independent. Rederive in this way for the quantity N above the Poisson distribution. 

Poisson distribution of mean n =0.0121 for events with NHIT 2 30. Hence the rate of occurrence of 6 events in a 10 second time interval due to a statistical fluctuation is less than one in 7 x 10 years in our experiment. 

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... 

It is useful to find a quantity that could serve us as a measure of these density fluctuations. Its simplest characteristic is the dispersion of a number of particles N in some volume V i.e., (N2) — (N)2. The distinctive feature of the classical ideal gas is a simple relation between the dispersion and macroscopic density (TV2) - (TV)2 = (IV) = nV. Moreover, all other fluctuation characteristics of the ideal gas, related to the quantity (Nm, could also be expressed through (TV) or density n. Therefore, in the model of ideal gas the density n is the only parameter characterizing the fluctuation spectrum. Such the particle distribution is called the Poisson distribution. It could be easily generalized for the many-component system, e.g., a mixture of two ideal gases. Each component is characterized here by its density, nA and nB density fluctuations of different components are statistically independent, (IVAIVB) = (Na)(Nb). 

Equation (1) is the most general and rigorous analytical result. However, it does not take account of the aggregation of similar defects and hence it is applicable only at not very large irradiation doses (up to a concentration of defects (l/2)no, where no is concentration at saturation). In fact, here the existence of clusters of similar defects is allowed, but it is actually assumed that these clusters are statistical fluctuations of the Poisson distribution of similar defects, which does not reflect a real pattern of cluster formation with a substantially non-Poisson spectrum of fluctuations. It is assumed implicitly in equation (1) that, after each event of creating a new pair of defects, the entire system of defects is stirred to attain the Poisson distribution. In the case of the absence of the defect correlation in genetic pairs we arrive at equation (2). 

The statistics of processes such as radioactive decay and emission of light that produce a flux of particles or distributive polymerase enzymes that add residues at random to growing polymer chains obey the Poisson distribution (see Chapter 14). The number of particles measured per unit time or the number of residues added to a particular chain varies about the mean value x according to equation 6.41. 

In recent times, the term superstatistics has been coined [153] to denote an approach to non-Poisson statistics, of any form, not only the Nutting (Tsallis) form, as in the original work of Beck [154]. We note that Cohen points out explicitly [153] that the time scale to change from a Poisson distribution to... 

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