Gamma distribution, continuous distributions

For a lucid discussion of the gamma distribution, see Chap. 17 in N. L. Johnson and S. Kotz, Continuous Univariate Distributions, Vol. 1, Wiley, New York, 1970. 

The continuous generalization of the case Xf = l/N would seem to be one where ) (x) is constant, and this seems to create a problem, as discussed later. Now consider the case where equals the gamma distribution [Pg.63]

The conceptual point is as follows. The discrete case X/ = l/N requires the mole fractions of all components to be equal to each other. That does not, however, correspond to a constant X (x) in the continuous description, because ) (x)dx is the mole fraction of species between x and x + dx, and one would need to require (x)dx to be constant. But this can only be done if one has chosen a specific scaling for the label x Any label x that is given by a monotonous function x (x) = X would be legitimate, and of course x)dx could be taken as constant for only one such scale. In other words, in a continuous description one has chosen some label x. The form of the mole fraction distribution must then satisfy certain constraints such as the one discussed earlier for the gamma distribution. The problem is related to the more general problem of the correct generalization to a continuous description of nonlinear formulas. 

To illustrate, consider a uniform cash flow of 1000 per year beginning at some uncertain time m and continuing for a duration of t years. The delay to initiation is uniformly distributed between 6 months and 1 year. The project duration is gamma distributed with mean of 3 years and standard deviation of 1 year the parameters of the gamma distribution yielding these statistics are a = 3 and b = 9. The nominal interest rate is 10% compounded continuously. It is assumed that the initiation time and project dmation are independent random variables. Our problem is to determine the equivalent present value of these cash flows. (This problem is taken from Park and Sharp-Bette (1990, p. 411)... 

We propose here to define the bivariate process a state dependent stochastic process similar to the one presented in Zouch et al. (2011). The evolution of degradation over a period of time T is given by positive increments for the degradation processes respectively (dp, A9) which are continuous random variables. A suitable candidate for these laws of distribution of each increment of the structure s degradation (Van Noortwijk, 2009) is the gamma distribution with two parameters (and... 

So what is a gamma-globulin molecule Could it not be a continuous system of molecular species, each slightly different from the next, wherein their properties would have a gaussian distribution The concept developed in this report shows how one could examine the properties of a system of closely related proteins in such a way that the molar ratio of certain amino acids, when related to mobility, exclude molecule species that do not belong to the same system. These parameters not only relate one molecular species to another, but possibly suggest their biosynthetic origin. 

Gamma-ray and a-ray spectrometry are important tools of nuclear and radiochemistry. They are mainly used for identification of radionuclides. Because of the continuous energy distribution of P radiation, yS-ray spectrometiy is less frequently applied. 

Five standard distributions for continuous r.v. s are uniform for model situations where aU outcomes over an interval [a, b] are equaUy hkely, normal playing an important role in explaining many real-life phenomena, exponential for a continuous counterpart of the geometric distribution used to model the time to an event, gamma obtained by summing n i.i.d. exponential r.v. s, and beta being a generahz-ation of the uniform distribution. 

FIGURE 22.3 (continued) Particle size distribution (diameter in nm) of Pd nanoparticles on MWCNTs with 0.07 M SDS irradiated at a dose rate of 10 kGy/h and different doses at (d) 40 kGy, (e) Plot of particle size vs. dose. (Reprinted from Radiat. Phys. Chem., 81 (1), Rojas, J.V. and Castano, C.H, Production of palladium nanoparticles supported on multiwalled carbon nanotubes by gamma irradiation, 16-21, Copyright 2012, with permission from Elsevier.)... 

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