Probability equations

For the special case that the system model is the sum of probabilities (equation 2.7-19) with equation 2 7- H gives the result equation 2.7-20, This says that (lie system mean is the sum of the means of he component distributions. 

Thus, when the attention of the mathematicians of the time turned to the description of overdetermined systems, such as we are dealing with here, it was natural for them to seek the desired solution in terms of probabilistic descriptions. They then defined the best fitting equation for an overdetermined set of data as being the most probable equation, or, in more formal terminology, the maximum likelihood equation. 

Under the proper conditions (said conditions being that the errors that prevent all the data relationships from being described by a single equation are normally [1, 2] distributed) it can be proven mathematically that the most probable equation is exactly the one that is the least square equation. While we have discussed this point... 

Further, for planes 1 and M the segment/surface interactions will necessitate the introduction of the Boltzmann factor in the probability equation. Thus ... 

When fragments (either primary or secondary) are ejected, it is often necessary to calculate the probability of their impacting a particular target. Work by Klein (20) and Hackett (21) gives the hit probability equation as ... 

The most appropriate models for CYC is probably equation (14) as this is the simplest and is most closely related to the results of the factor analysis which indicated that the variances in CYC were most closely related to those in V and PB. The inclusion of coarse particle manganese as a soil tracer diminished the significance of the coefficient of PB and the contribution of automotive sources. Ideally, MN would be used as a tracer for resuspended soil but Interferences from the use of MMT as a fuel additive during part of the period in which this data were collected make this a mixed source tracer for the contributions of automobiles and soil resuspension. 

This is merely the hydrodynamic continuity equation p = — div pv written for an arbitrary number of dimensions.If one now lets y(t) run over all realizations of Y(t), with their appropriate probabilities, equation (5.2) becomes a linear stochastic differential equation for p(w, t). 

Three widely used distribution equations, discussed by Bevans (JL), include the Rosin-Rammler (7L) and Nukiyama-Tanasawa (6L) equations as well as the log-probability equation. A fourth relationship, the upper-limit equation of Mugele and Evans (5L), is also discussed. Hawthorne and Stange also discuss the Rosin-Rammler relationship (4L, 8L). An excellent analysis of distributions is given by Dubrow (SL), who has studied atomized magnesium powders. 

The standard deviations may also be obtained easily from the grids based on relationships derivable from the probability equation. Thus for arithmetic-probability plots... 

Following Ref. 4, the polymer size distribution in the Smith-Ewart limit D/B f oo can be developed from the probability equations ... 

Get the heck out of here, you might say, how did you get that Figure it out But, this relationship is certainly correct. Just use Equations 6-10 and 6-51, substitute into Equation 6-53 and you will get the definition of x (Equation 6-45). Going back to the expression in Equation 6-53 and substituting for the conditional probabilities (Equations 648 to 6-50), we then obtain an expression for in terms of the reactivity ratios and the molar feed ratio (Equation 6-54) ... 

Just as in the derivation of the copolymer equation for the terminal model, we start with a reversibility relationship P3 AAB = P3 BAA. Now we must use second-order Markovian statistics to write this in terms of conditional probabilities (Equation 6-64) ... 

The probability arguments we have reproduced on the preceding pages can also be applied to describe the sequence of isomers in polymers like atactic polypropylene, or the placement of cis-1,4 and trans-1,4 units in diene polymers. But if we write one more probability equation you will probably hurl this book from the highest tower you can... 

To apply Equation (1.14), recorded data are arranged into arrays either from the highest to the lowest or from the lowest to the highest. The number of values above a given element and including the element is counted and the probability equation applied to each individual element of the array. Because the number of values above and at a particular element is a sum, this application of the equation is, in effect, an application of the probability of the union of events. The probability is called cumulative, or union probability. After all union probabilities are calculated, an array of probability distribution results. This method is therefore called probability distribution analysis. This method will be illustrated in the next example. 

For a normal distribution of variance, the spread of data about the mean is described by the probability equation... 

Table 2.9 Tabular solution to the normal probability equation...

This is a probable equation. The stoichiometry has not yet been uniquely proved. 

According to Eq. (50), each resonance state contributes independently to the survival probability. Equation (50) can be shown to provide an appropriate approximation even for NO2 [HO] in which the resonances overlap. An interesting discussion of a more general decay function for overlapping resonances can be found in Ref. 274. 

The last variable to be included in the fault seal probability equation is the net to gross ratio. The assumptions used in the formulation of the equation are that with low net to gross ratios, say <0.4, there is a greater likelihood for sealing due to shale smear than in higher net to gross ratio successions (see also Bou-vier et al., 1989 Harding and Tuminas, 1989). 

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