Distribution functions Poisson

We propose to describe the distribution of the number of fronts crossing x by the Poisson distribution function, discussed in Sec. 1.9. This probability distribution function describes the probability P(F) of a specific number of fronts F in terms of that number and the average number F as follows [Eq. (1.38)] ... 

Resolving power R is a measure of specificity and is a primary factor contributing to the probability of false positives. The salient issue is the probability that a background signal will overlap with a target explosives signal. We can qualitatively express this probability using a Poisson distribution function of the form... 

The olefin product distribution has been calculated by assuming an aluminum buildup according to the Poisson distribution function. For these calculations, it was assumed that the alkylation reaction is conducted to 58% completeness and that two ethylene groups per pass are added to the alkylaluminums during buildup. 

This equation with a fixed value of N and a variable value of t is not a Poisson distribution function (in contrast to Eq. 6.3), but a gamma density function [14], with a first moment given by... 

In the case of addition polymerization without termination, the number fraction distribution function (the probability that a given chain has degree of polymerization N) is given by the Poisson distribution function ... 

Among a large number of small polymer droplets, each of volume v, the fraction of droplets which contain exactly z heterogeneities of the kind "A" that usually induce crystallization follows a Poisson distribution function (45) ... 

Figure 19.16B shows a typical Poisson distribution function. A very important distinction between this function and the normal distribution function is that, in order to characterize the latter, we must know both p. and a because the broadness of the normal distribution is independent of its mean. In contrast, the Poisson distribution curve is completely characterized by its mean alone. The broadness of the distribution is a function of the mean, and is given by... 

Equation (3.2) represents the Poisson distribution and/(x) is the Poisson distribution function. Like binomial distribution, Poisson distribution is not continuous. To calculate the average (mean) value of x, we must use the operator summation. The value of m that is the product of Np plays a characteristic role in Poisson distribution. An important assumption is that the distribution f x) is through the area with uniform probability. In comparison, the binomial distribution has a much wider spread from the lowest value to the highest value than the Poisson distribution. 

The curve in Figure ll.ld is a plot of a simulated distribution of a theoretical polymer produced by an ideal living polymerization (Zhou, 2002). The plot is generated by a Poisson distribution function (Flory, 1940). The polymer distribution (FDMS) produced by the polyhomologation reaction closely approximates the Poisson distribution. This is strong support for the living nature of the polyhomologation reaction. 

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