Beta density function

Figure 1. Beta density functions. Reproduced with permission from Ref. 20. Copyright 1970, Houghton Mifflin Co..

The percentage of impurities per batch in a certain chemical product is a random variable Y having beta density function with parameters a = 0.5 and b = 10. A batch with more than 10% impurities will be rejected. What is the probability that a randomly selected batch will be rejected because of excessive impurities ... 

Wall, C., B. J. Boersma, and R Moin (2000). An evaluation of the assumed beta probability density function subgrid-scale model for large eddy simulation of nonpremixed, turbulent combustion with heat release. Physics of Fluids 12, 2522-2529. 

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

Figure 4.5 Beta probability density functions for different parameter pairs oti and a2.

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... 

FIGURE 5.8 Shapes of commonly used probability density function, (a) Double delta function, (b) Beta function. 

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 ... 

The fourth step, which computes the density, is one of the algorithm s most expensive. Figure 2 contains an efficient loop structure for the construction of the alpha density and its gradient at all grid points on the sphere (the beta density and its gradient are evaluated simultaneously in exactly the same manner). It should be noted that the innermost loop, whose cost is 0(h2N), contains only a SAXPY operation and does not involve basis function gradients. The density and... 

Xu, X.P, Case, D.A. Probing multiple effects on 15N, 13C alpha, 13C beta, and 13C chemical shifts in peptides using density functional theory. Biopolymers 2002, 65, 408-23. 

Fitting a subjective belief form requires that the questions be posed in terms of statistical parameters. That is, decision makers could be asked to first consider their uncertainty regarding the true value of a given probability and then estimate their mean, mode, or median belief. This approach can be further extended by asking decision makers to describe how certeiin they are of their estimate. For example, a worker might subjectively estimate the mean and veuiance of the proportion of defective circuit boards before inspecting a small sample of circuit boards. If the best estimate corresponds to a mean, mode, or median, and the estimate of certainty to a confidence interval or stemdard deviation, a functional form such as the Beta-1 probability density function (pdf) can then be used to fit a subjective probability distribution (Clemen 1996 Buck 1989). 

The bisection method (Raiffa 1968) is another direct technique for attempting to estimate a subjective probability density function (pdf). This technique is somewhat more general than fitting the subject s belief with a functional form, such as the beta-1, since it makes no parametric assumptions. The bisection method involves two steps which arc repeated imtil the subject s belief is adequately described. Following this approach, the first step is to determine the median (Pos) of the subjective pdf. This question is posed to the decision maker in a form such as For what value of p do you feel it is equally likely the true value pt is greater than or less than p This step is then repeated for subintervals to obtain the desired level of detail. 

R. M. Osuna, R. P. Ortiz, V. Hernandez, J. T. L. Navarrete, M. Miyasaka, S. Rajca, A. Rajca and R. Glaser, Helically annelated and cross-conjugated beta-ohgothiophenes a Fourier transform Raman spectroscopic and quantum chemical density functional theory study, J. Phys. Chem. C, 111, 4854-4860 (2007). 

Probability density function where B is the beta function. ... 

The functions betapdf(x,a,b) and betacdf(x,a,b) are available in the MATLAB Statistics Toolbox for calculating beta probability density function and beta cumulative distribution function. 

The relative humidity, Y, when measured at a certain location, can be described by the beta probability density function with parameters a = 4 and b = 3. Find the probability that humidity lies between 40%... 

Bpdf( ) = Beta probability density function (PDF). a and b = Parameters for the Beta distribution of the sample damage ratios for a GM intensity bin. 

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