Poisson statistic

The integration of Eq. (6.106) is central to the kinetic proof that living polymers follow Poisson statistics. The solution of this differential equatior is illustrated in the following example. 

The generation of photons obeys Poisson statistics where the variance is N and the deviation or noise is. The noise spectral density, N/, is obtained by a Fourier transform of the deviation yielding the following at sampling frequency,... 

Consequently, consider an infinite one-dimensional line or rod along which fractures occur randomly with an average frequency of Nq per unit length as illustrated in Fig. 8.19. Randomly distributed points on an infinite line obey Poisson statistics and the probability of finding n fractures in a length, /, is given by... 

Figure 8.20. Cumulative number distribution of fragments from four expanding ring experiments (10 fragments each) and comparison with one-dimensional theoretical distribution based on Poisson statistics.

DECODING ID AND 2D MULTICOMPONENT SEPARATIONS BY USING THE SMO POISSON STATISTICS... 

The eigenvalues of SG are here characterised in terms of two quantum numbers, an angular momentum quantum number m and a second quantum number counting the eigenvalues in each m manifold. If the spectra for different m are uncorrelated, one expects Poisson statistics of the total spectrum in the limit n — oo. 

The central assumption of ion counting is that ions arrive at the detector at random, i.e., that the probability of arrival of an ion is the same for any time interval of a same length. The number , of ions i arriving at any collection device during the time interval dt is therefore subject to Poisson statistics , is proportional to 5t, the average count rate is n/8t, and its... 

If each cell responds to radiation autonomously (i.e., independently of any other cell), an upper limit of effect probability E D) is the probability that at least one energy deposition event actually occurred in it. This latter is given [by Poisson statistics, see Eqs. (13) and (14)] by 1—exp(—Z)/zf) and thus, for any dose, one must have ... 

The content uniformity which can be obtained depends, according to established theoretical considerations, on the particle size of the active substance (19,20). As a rough estimate for the obtainable relative standard deviation 5 rei of the content of the active substance, the following rule can be applied, based on Poisson statistics ... 

In the experiment, the transmission intensities for the excited and the dark sample are determined by the number of x-ray photons (/t) recorded on the detector behind the sample, and we typically accumulate for several pump-probe shots. In the absence of external noise sources the accuracy of such a measurement is governed by the shot noise distribution, which is given by Poisson statistics of the transmitted pulse intensity. Indeed, we have demonstrated that we can suppress the majority of electronic noise in experiment, which validates this rather idealistic treatment [13,14]. Applying the error propagation formula to eq. (1) then delivers the experimental noise of the measurement, and we can thus calculate the signal-to-noise ratio S/N as a function of the input parameters. Most important is hereby the sample concentration nsam at the chosen sample thickness d. Via the occasionally very different absorption cross sections in the optical (pump) and the x-ray (probe) domains it will determine the fraction of excited state species as a function of laser fluence. 

A summary of the instrumental variables (that may be optimized by the designer or experimenter) and their relations with the data-averaging variables is given in Table V. These relations are based on Poisson statistics. 

In reading various HO measurement reports [including Hard et al. (78)], we find too little attention given to clearly expressing how the uncertainty limits are derived. This negligence may be the result of assuming such uncertainties are trivial to calculate. However, future reports should state explicitly whether la, 2a, 90% confidence limits, and so forth employ Gaussian or Poisson statistics and whether the quoted uncertainties are internal to the ambient HO data or include calibration uncertainties. 

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... 

Consider recrystallization and grain growth in an infinite thin sheet. Assume that the nucleation rate of recrystallized grains is a linear function of temperature above a critical temperature, Tc, and the nucleation rate is zero for T grain-growth rate, R, is constant and independent of temperature. Suppose that at time t = 0 the sheet is heated at the constant rate T(t) = Tc/2 + /3t. Using Poisson statistics, the probability that exactly zero events occur in a time t is p0 = exp(— Nc))-... 

To avoid bulky calculations, we restrict ourselves by the following problem statement particles A and B have equal diffusion coefficients, DA = Db = D, fluctuating particle sources in [Pg.90]

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