Beta distribution

An informative conjugate prior distribution can be formulated in tenns of a beta distribution ... 

We can think of the beta distribution as the likelihood of a prior successes and (3 failures out of a + (3 experiments. The F functions in front serve as a normalization constant, so that li/>(0) dQ =. Note that for an integer, Y x + ) = xl The posterior distribution that results from multiplying together the right-hand sides of Eqs. (2) and (3) is also a beta distribution ... 

It is tempting to use Eq. (7) to derive Eq. (6), because they have similar forms given the relationship of the F function to the factorial. But the binomial and the beta distribution are not normalized in the same way. The beta is nonnalized over the values of 0, whereas the binomial is nonnalized over the counts, y given n. That is. 

This table indicates that if a beta function prior is convoluted with a binomially distributed update, the combination (the posterior) also is beta distributed. 

Maintenance data are treated by computing an average maintenance unavailability for each component type or system and fitting the data to a beta distribution. This is because maintenance outages are logged on a system basis in many cases. 

P-diketone chelates, 25 89-91 Beta distribution, 26 1020 P-eleostearic acid, physical properties,... 

A random variable X is distributed as the beta distribution over the range [0,1] if its density function is given by... 

No simple form of the moment generating function exists. In the special case where 0C =a2 = 1, the beta distribution reduces to the uniform distribution over [0, 13- Finally, we will frequently refer to Snedecor s F-distribution. A random variable defined over ]0, + 00 [ is distributed with the F-distribution with v, and v2 degrees of freedom... 

The probability density distribution fx(x) of the random variable A- is a beta distribution with parameters m and n. Therefore... 

From equation (4.1.32), the parameters m and n of a beta distribution can be computed from the mean and the variance as... 

The most familiar estimation procedure is to assume that the population mean and variance are equal to the sample mean and variance. More generally, the method of moments (MOM) approach is to equate sample moments (mean, variance, skewness, and kurtosis) to the corresponding population. Software such as Crystal Ball (Oracle Corporation, Redwood Shores, CA) uses MOM to fit the gamma and beta distributions (see also Johnson et al. 1994). Use of higher moments is exemplified by fitting of the... 

Supposing that one has decided on bounds for a variable, one can fit a distribution that has a bounded support, such as the beta distribution or Johnson SB distribution. Alternatively, in a Monte Carlo implementation, one may sample the unbounded distribution and discard values that fall beyond the bounds. However, then a source of some discomfort is that the parameters of the distribution truncated in this way may deviate from the specification of the distribution (e.g., the mean and variance will be modified by truncation). It seems reasonable for Monte Carlo software to report the percentage discarded, and report means and variances of the distributions as truncated, for comparison to means and variances specified. 

The classes specified in a robust Bayesian analysis can be defined in a variety of ways, depending on the nature of the analyst s uncertainty. For instance, one could specify parametric classes of distributions in one of the conjugate families (e.g., all the beta distributions having parameters in certain ranges). Alternatively, one could specify parametric classes of distributions but not take advantage of the conjugacies. 

For the random variable in Exercise 9, what is the probability distribution of the random variable y = exl What is Ely Prove that the distribution of this y is a special case of the beta distribution in (3-40). 

The probability density function, written as pif), describes the fraction of time that the fluctuating variable/ takes on a value between/ and/ + A/. The concept is illustrated in Fig. 5.7. The fluctuating values off are shown on the right side while p(f) is shown on the left side. The shape of the PDF depends on the nature of the turbulent fluctuations of/. Several different mathematical functions have been proposed to express the PDF. In presumed PDF methods, these different mathematical functions, such as clipped normal distribution, spiked distribution, double delta function and beta distribution, are assumed to represent the fluctuations in reactive mixing. The latter two are among the more popular distributions and are shown in Fig. 5.8. The double delta function is most readily computed, while the beta function is considered to be a better representation of experimentally observed PDF. The shape of these functions depends solely on the mean mixture fraction and its variance. The beta function is given as... 

If the observation vector x is not independent of the estimators x and S, but is included in their computation, then follows a Beta distribution with m/2 and n — m— l)/2 degrees of freedom [190] ... 

Univariate EQMOM can be applied to an NDE defined on a finite interval (Yuan et at., 2012). Eor example, if the NDE is nonzero only on the interval [0,1], then we can define the kernel density functions using a beta distribution ... 

Most daughter distribution functions can be easily extended to bivariate problems. Let us consider two examples. In the first example particles with two components A and B are described. The particulate system is defined in terms of the size of these particles dp and the composition of the particles 0, expressed for example as the mass fraction of component A in the particle. When a particle breaks we can assume for example that the amount of component A is partitioned among the daughters proportionally to the mass of the fragments. Under these hypotheses, and the additional assumption of binary breakage following the beta distribution, the resulting bivariate distribution is... 

Beta distribution parameters cross-sectional area... 

The third source arises from the extraction efficiency for the removal of solute from water, a technique common to most methods. Peters (6), has shown this to introduce another negative bias within 8 percent therefore a beta distribution with a 95-percent confidence level of -8 percent was selected. Finally, the accuracy of the detection method was modeled by a normal distribution with a 95-percent confidence level of +2 percent as recommended by Mallon and Harrison ( 7). 

The column method also has several steps which may result in a negative bias on measured aqueous solubility. The lack of water-solute equilibrium in the generator column itself may produce an outlet concentration lower than the "true solubility according to Stolzenburg and Andren (8). This result also has been modeled with a beta distribution (93-percent confidence level of -3 percent). 

The efficiencies for adsorption and extraction steps again are estimated to be similar to those suggested by Peters (6), therefore, the same beta distributions as before are employed. However, note that for the column generator, two-step organic removal typically is used. Finally, one may ascribe the same detection limits as those obtained for the shake-flask method. 

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