Factorial cumulant

Express the first few in terms of the moments. Show that the Poisson distribution (2.10) is characterized by the vanishing of all factorial cumulants beyond 0X. Exercise. Find the factorial moments and cumulants of (1.5). 

Exercise. Ordinary cumulants are adapted to the Gaussian distribution and factorial cumulants to the Poisson distribution. Other cumulants can be defined that are adapted to other distributions. For instance, define the %m by... 

Multivariate factorial cumulants, indicated by square brackets, are... 

Exercise. The factorial cumulant of the sum of two statistically independent variables is the sum of their factorial cumulants. A factorial cumulant involving two mutually independent sets of variables vanishes. 

Exercise. Show that the factorial cumulants (1.3.13) for a process Y are given by... 

Exercise. Find the factorial cumulants (1.2.17) for the decay process. 

Exercise. It is also possible to deduce from (6.4) equations for the second moments. The most condensed way of expressing the result is in terms of the factorial cumulants (1.3.13) ... 

In the same way one finds by laborious algebra an equation for the second moments , which simplifies when expressed in the factorial cumulants... 

To rewrite (2.2) in the same way we first have to define the factorial cumulant in the continuous description. Divide (2.3) by A2 and replace a, with r1,r2 ... 

As the factorial cumulant must remain finite both at large and at small distance the only admissible solution is g = 0. The covariance of the fluctuations is therefore... 

For the factorial cumulant this last equation gives... 

In the diffusion case it was essential to add the transfer terms to the factorial cumulant rather than to the variance in order to do justice to the noisy character of diffusion, but in the present case it makes no difference. We merely use the factorial cumulant because it makes the equations simpler. 

The last two lines consist of a mass of terms, which can be reduced by introducing the factorial cumulants... 

It is less forbidding than it looks. The first three lines are the linearized Boltzmann operator acting on the factor u(rin the factorial cumulant the next three lines are the same operator acting on the factor u(r2,p2) lines seven and eight represent the sources of the fluctuations and on the last line the flow terms have been added for both factors. The equation has therefore the general form (VIII.6.8), when A is identified with the linearized Boltzmann operator including the streaming term. 

In that case the seventh and eighth lines vanish, see Exercise below. Also the factorial cumulant will be independent of time, and we write it as a matrix <9,... 

Exercise. Derive in analogy to (4.7) for the factorial cumulant of the occupation number... 

Exercise. The factorial cumulants are suitable for classical particles, but for bosons the generalized cumulants nm defined in (1.2.20) are better suited. Express (4.8) in terms of n2-... 

A retrospective cohort study was done in 1982-83 on 28,460 benzene workers in China, all of whom had worked in various factories for at least half a year between 1972 and 1981 however, exposure and employment duration were not necessarily limited to those years (Yin et al. 1987a). Thirty cases of leukemia (23 AML, 7 CML) with a mean latency of 11.4 years (0.8-49.5 years) were found in the benzene cohort, as opposed to 4 cases in a matched control cohort. Twenty-five of the leukemic workers had already died. Information on exposure levels was collected from company records, but there is no indication of the extent of these records with the exception of a note that three of the reported levels were based on only one measurement. Mean benzene levels to which workers with leukemia were exposed ranged from 3 to 313 ppm, with the majority falling between 16 and 157 ppm. Exposure ranges from which the means were derived were rather wide, indicating the possibility of at least occasional high exposures. Only 4 upper-level measurements were less than 10 ppm, while half of the remaining measurements were between 10 and 100 ppm, and the other half were between 100 and 2,000 ppm. The authors observed that the cumulative mortality due to leukemia was proportional to the duration of the exposure to benzene for exposure of up to 20 years and then leveled off. 

Using a predictive model developed from mesothelioma data from studies of asbestos insulation workers (Peto et al. 1982), asbestos textile workers (Peto 1980), amosite factory workers (Seidman 1984), and asbestos-cement workers (Finkelstein 1983), EPA (1986a) estimated that continuous lifetime exposure to air containing 0.0001 f/mL of asbestos would result in about 2-3 cases of mesothelioma per 100,000 persons. The corresponding cumulative lifetime exposures associated with excess risks of 10 " -10 are shown in Figure 3-1. Cumulative exposure levels of 0.031, 0.0031, 0.00031, and 0.000031 f-yr/mL represent excess mesothelioma risks of 10" , 10 , 10, and 10 ", respectively. Appendix D provides further details on the derivation of these risk estimates. Currently (in 2001), EPA is in the process of reviewing their cancer risk estimates for asbestos fibers. 

The analysis by means of normal probabflify plots is an alternative technique for trying to find out, among the results calciflated for a factorial design, those that correspond to real effects, as opposed to those caused only by noise. Its working principle is based on the idea of cumulative probabflify, which was introduced in Exercise 2.9. We shall start by discussing this concept in more detail. 

Correspondence between the calculated effects of the 2 factorial design and their cumulative probability values... 

Equations can also be derived for the factorial cumulants defined by the relation... 

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