Poisson approximation

The proof that these expressions are equivalent to Eq. (1.35) under suitable conditions is found in statistics textbooks. We shall have occasion to use the Poisson approximation to the binomial in discussing crystallization of polymers in Chap. 4, and the distribution of molecular weights of certain polymers in Chap. 6. The normal distribution is the familiar bell-shaped distribution that is known in academic circles as the curve. We shall use it in discussing diffusion in Chap. 9. 

Waterman, M. S., Vingron, M. (1994) Sequence comparison significance and Poisson approximation, Statistical Science 9 401-418. 

Given these assumptions, the distribution of the observed total number of counts according to probability theory should be binomial with parameters N and p. Because p is so small, this binomial distribution is approximated very well by the Poisson distribution with parameter Np, which has a mean of Np, and a standard deviation of Np. The mean and variance of a Poisson distribution are numerically equal so, a single counting measurement provides an estimate of the mean of the distribution Np and its square root is an estimate of the standard deviation /Np. When this Poisson approximation is valid, one may estimate the standard uncertainty of the counting measurement without repeating the measurement (a Type B evaluation of uncertainty). 

Although all the assumptions stated above are needed to ensure that the distribution of the total count is binomial, not all the assumptions are needed to ensure that the Poisson approximation is valid. In particular, if the source contains several long-lived radionuclides, or if long-lived radionuclides are present in the background, but all atoms decay and produce counts independently of each other, and no atom can produce more than one count, then the Poisson approximation is still useful, and the standard deviation of the total count is approximately the square root of the mean. 

Example 2.21 Poisson Approximation to Binomial from Tooth Cavities. In Example 2.17, let us see how well the Poisson distribution approximates the binomial distribution for the tooth cavity problem even though n = 20 is not very large and p = 0.10 is not very small. 

Poisson approximation. For small enough values of p and large enough values of (p < 0.1 and > 20 will do) the binomial distribution can be approximated by a Poisson distribution with parameter (expected value) p = np (see O Figs. 9.5 and O 9.7) ... 

Nuclear spectra. Before sketching out the solution for the fitting problem in general, a special class of spectra called nuclear spectra will be considered, in the case of which the Y)s are counts, for which the Poisson approximation and the normal approximation equally hold. In other words, for the spectra considered, the Y,-values have N(/i [Pg.449]

Too small (i.e., too good) a can be obtained, when the dead time is so large that the Poisson approximation built into the merit function ofOEq. (9.144) overestimates the variance in the denominator. In such a case, a more realistic result is obtained, if — on the basis of Eq. (9.111) - the calculated x -value is divided by (1 — Ovg), where 0Vg is the percentage of dead time divided by 100. 

Numerical evaluation of the numerator of Eq. (7) reduces to a time-invariant two-component parallel system reliability analysis. It is clear that the first part of Eq. (5) represents the building block for the solution of both time-invariant and time-variant reliability problems (Der Kiureghian 1996). Using Eq. (7), Poisson approximation to the failure probability, Pf oisson(T), is obtained as (under the hypothesis that... 

Marmur [12] has presented a guide to the appropriate choice of approximate solution to the Poisson-Boltzmann equation (Eq. V-5) for planar surfaces in an asymmetrical electrolyte. The solution to the Poisson-Boltzmann equation around a spherical charged particle is very important to colloid science. Explicit solutions cannot be obtained but there are extensive tabulations, known as the LOW tables [13]. For small values of o, an approximate equation is [9, 14]... 

This may be solved numerically or within some analytic approximation. The Poisson equation is used for obtaining the electrostatic properties of molecules. 

Magnesium alloys have a Young s modulus of elasticity of approximately 45 GPa (6.5 x 10 psi). The modulus of rigidity or modulus of shear is 17 GPa (2.4 X 10 psi) and Poisson s ratio is 0.35. Poisson s ratio is the ratio of transverse contracting strain to the elongation strain when a rod is stretched by forces at its ends parallel to the rod s axis. 

Mathematical Models for Distribution Curves Mathematical models have been developed to fit the various distribution cur ves. It is most unlikely that any frequency distribution cur ve obtained in practice will exactly fit a cur ve plotted from any of these mathematical models. Nevertheless, the approximations are extremely useful, particularly in view of the inherent inaccuracies of practical data. The most common are the binomial, Poisson, and normal, or gaussian, distributions. 

When the value of p is very close to zero in Eq. (9-77), so that the occurrence of the event is rare, the binomial distribution can be approximated by the Poisson distribution with X = np when n > 50 while npi < 5. 

SASA), a concept introduced by Lee and Richards [9], and the electrostatic free energy contribution on the basis of the Poisson-Boltzmann (PB) equation of macroscopic electrostatics, an idea that goes back to Born [10], Debye and Htickel [11], Kirkwood [12], and Onsager [13]. The combination of these two approximations forms the SASA/PB implicit solvent model. In the next section we analyze the microscopic significance of the nonpolar and electrostatic free energy contributions and describe the SASA/PB implicit solvent model. 

The continuum electrostatic approximation is based on the assumption that the solvent polarization density of the solvent at a position r in space is linearly related to the total local electric field at that position. The Poisson equation for macroscopic continuum media... 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

Radiolytic oxidation alters most of the important properties of graphite, including strength, elastic modulus, work of fracture, thermal conductivity, permeability, and diffusivity but does not affect the thermal expansion coefficient or Poisson s ratio. The effects of radiolytic oxidation on the properties of a wide range of graphites have been studied in the U.K. [7,73,74] where it was found that, to a first approximation, they can be described by similar relationships ... 

If we take the critical stress as the yield stress then for many plastics, the ratio of adE is approximately 35 x 10 2. Using Poisson s ratio, v = 0.35 and taking p = 0.6, as before, then... 

Hull also proposed that the shear modulus and Poisson s Ratio for a random short fibre composite could be approximated by... 

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