Gamma distributions

For exponential distribution, the prior distribution of the failure rate X is usually characterized by its conjugate distribution, Gamma distribution (Zhang Jiang 2000, Wang 1993) whose probability density function is. 

Radiometric ore sorting has been used successfully for some uranium ores because uranium minerals emit gamma rays which may be detected by a scintillation counter (2). In this appHcation, the distribution of uranium is such that a large fraction of the ore containing less than some specified cut-off grade can be discarded with tittle loss of uranium values. Radioactivity can also be induced in certain minerals, eg, boron and beryllium ores, by bombarding with neutrons or gamma rays. 

Distribution models are curvefits of empirical RTDs. The Gaussian distribution is a one-parameter function based on the statistical rule with that name. The Erlang and gamma models are based on the concept of the multistage CSTR. RTD curves often can be well fitted by ratios of polynomials of the time. 

Figure 8-38 shows the residenee time distributions of some eom-mereial and fixed bed reaetors. These shapes ean be eompared with some statistieal distributions, namely the Gamma (or Erlang) and the Gaussian distribution funetions. However, these distributions are represented by limited parameters that define the asymmetry, the peak. 

An Excel spreadsheet (Example8-7.xls) was used to determine the various RTD functions and the computer program PROGS 1 was used to simulate the model response curve with the experimental data. The results show the equivalent number of ideally mixed stages (nCSTRs) for the RTD is 13.2. The Gamma distribution function from Equation 8-143 is ... 

Similarly, prior information distributed as a gamma lunctiou with an exponentially distributed update gives a posterior that also is gamma distributed. 

The Bayes conjugate is the gamma prior distribution (equation 2.6-11). When equations 2.6-9 and... 

A number of issues arise in using the available data to estimate (he rates of location-dependent fire occurrence. These include the possible reduction in the frequency of fires due to increased awareness. Apostolakis and Kazarians (1980) use the data of Table 5.2-1 and Bayesian analysis to obtain the results in Table 5.2-2 using conjugate priors (Section 2.6.2), Since the data of Table 5.2-1 are binomially distributed, a gamma prior is used, with a and P being the parameters of the gamma prior as presented inspection 2.6.3.2. For example, in the cable- spreading room fromTable 5.2-2, the values of a and p (0.182 and 0.96) yield a mean frequency of 0.21, while the posterior distribution a and p (2.182 and 302,26) yields a mean frequency of 0.0072. 

The magnitude of CV is normally ealeulated numerieally, although graphieal methods are also possible. Exeeptionally, CV ean be evaluated analytieally for a defined distribution funetion (see Randolph and Larson, 1988). The gamma distribution is espeeially important in erystallization, and is eonsidered in more detail in Chapter 3. 

The mass distribution from the idealized MSMPR crystallizer is thus a Gamma function, as shown in Figure 3.8b. 

There are other distributions that can be used in a variety of reliability models. The Poisson, the extreme value, gamma, binomial, and Rayleigh distributions are sometimes used in specialized models. 

Half-life is the time taken to decrease the concentration of a drug to one-half its original value. There may be several phases in the elimination, and the most common is the so-called beta-phase. Alpha-phase is a distribution phase and gamma-phase is the terminal phase when the drug is finally leaving the tissues. 

The term vitamin E describes a family of eight antioxidants, four tocopherols, alpha (a), beta ((3), gamma (y) and delta (8), and four tocotrienols (also a, (3, y, and 8). a-Tocopherol is present in nature in only one form, RRR a-tocopherol. The chemical synthesis of a-tocopherol results in eight different forms (SRR, SSR, SRS, SSS, RSR, RRS, RSS, RRR), only one of which is RRR a-tocopherol. These forms differ in that they can be right (R) or left (S) at three different places in the a-tocopherol molecule. RRR a-tocopherol is the only form of vitamin E that is actively maintained in the human body and is therefore the form of vitamin E found in the largest quantities in the blood and tissue. A protein synthesized in the liver (a-TTP alpha-tocopherol transfer protein) preferentially selects the natural form of vitamin E (RRR a-tocopherol) for distribution to the tissues. However, the mechanisms for the regulation of vitamin E in tissues are not known... 

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