Beta function

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

In Equation 8, V is the value to be tested. Xv is the mean, and n is the number of values. The t-score is converted to a probability by a subroutine which calculates the incomplete Beta function. This could also be accomplished using an internal look-up table. 

In developing series solutions of differential equations and in other formal calculations it is often convenient to make use of properties of gamma and beta functions. The integral... 

Note, t — mixing time Q — volumetric flow rate V = volume of system 0 — V/q mean circulation time q = volumetric recirculation rate V[a. b) = incomplete beta function. 

Reinmuth [465] arrived at a solution for the Langmuir isotherm in the form of a series, involving the beta function. Levich et al. [362] had an approximate solution for a general adsorption isotherm. 

FIGURE 10.2. Various shapes of beta-function densities for the mixture fraction (from N. Peters). 

Ross (R15) and Ross and Curl (R16) assumed the /3(u, a) term to be given by the beta function, whereas Shiloh et al. (S21) and Delichatsios and Probstein (D5) assumed that a drop breakage results in two equal daughter drops. 

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. 

This beta function is zero in the absence of dipolar interactions. It vanishes for interactions linear in the spin operator and has nonzero values only for bilinear interactions. Furthermore, it is zero for Je approaching zero, so that a nonzero signal indicates residual anisotropic interactions, and it is free of signal attenuation by relaxation. The shape of the beta function has been shown to depend strongly on the strain in rubber samples [Cal3]. 

In this section, we present results found with the scalar-conditioned velocity algorithm introduced in the previous section using the NDF representation in Eq. (8.108). All of these calculations have been done with beta EQMOM using n = 2 nodes for the first quadrature and M = 50 nodes for the second. The initial NDF, where f e [0,1] is the particle volume, has one of two forms. In the first case, referred to as small spread, the NDF is represented by two beta functions, one near = 1/3 and one near f = 2/3, with a small spread parameter cr. The initial moments for this case are shown in Figure 8.28. As can be observed from this figure, the moments are nonzero only within the spatial domain x (-0.8, -0.2). Some statistics computed from the initial NDF are given in Figure 8.29. Note that the initial conditional velocity u(0,x,f) = 1 is independent of the particle size, and the mean and standard deviation of the volume do not depend on x. 

Beta functions" have been used to represent several sources of error. The beta function is extremely flexible in the sense that it can assume a great variety of shapes depending on values chosen for exponents, as seen in Figure 1, and seems particularly useful in representing negative bias. 

It can be simplified with Euler s Beta function B z,w). From its definition... 

Prom Eq. (45), the emission spectrum (3) follows. Finally, using integral representation for the Beta-function in Eq. (42) we arrive at... 

Other quantities of interest are the so-called beta-functions in the horizontal and vertical where... 

Divergence angle of synchrotron radiation emitted (including electron divergence and opening angle, y-1, contribution) Machine emittance Machine beta-function (in context)... 

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