Useful Probability Functions

A summary of useful probability functions is given in Table 7.6. 

---

The normal (Gaussian) distribution is the most frequently used probability function and is given by... 

Therefore, probability functions describing the lifetime distribution have to be defined they allow the future behavior of devices to be calculated based on data from experiments performed on a hmited population. Table 5.9.2 bsts the most frequently used probability functions for reliability calculations and gives brief definitions, as well as the range of values and their principle behavior over time. 

In the next section we shall adapt this probability function to the description of a three-dimensional coil. We conclude this section by noting that Eq. (1.21) may be approximated by two other functions which are used elsewhere in this book. For these general relationships we define v to be the number of successes-that is, some specified outcome such as tossing a head-out of n tries and define p as the probability of success in a single try. In this amended notation, Eq. (1.21) becomes... 

We desire to use the probability function derived above, so we recognize that the mass contribution of the volume element located a distance r from an axis through the center of mass is the product of the mass of a chain unit mp times the probability of a chain unit at that location as given by Eq. (1.44). For this purpose, however, it is not the distance from the chain end that matters but, rather, the distance from the center of mass. Therefore we temporarily identify the jth repeat unit as the center of mass and use the index k to count outward toward the chain ends from j. On this basis, Eq. (1.49) may be written as... 

The Poisson distribution can be used to determine probabilities for discrete random variables where the random variable is the number of times that an event occurs in a single trial (unit of lime, space, etc.). The probability function for a Poisson random variable is... 

The first step in data analysis is the selection of the best filling probability function, often beginning with a graphical analysis of the frequency histogram. Moment ratios and moment-ratio diagrams (with p as abscissa and as ordinate) are useful since probability functions of known distributions have characteristic values of p, and p. ... 

Frequency analysis is an alternative to moment-ratio analysis in selecting a representative function. Probability paper (see Figure 1-59 for an example) is available for each distribution, and the function is presented as a cumulative probability function. If the data sample has the same distribution function as the function used to scale the paper, the data will plot as a straight line. 

The procedure is to fit the population frequency curve as a straight line using the sample moments and parameters of the proposed probability function. The data are then plotted by ordering the data from the largest event to the smallest and using the rank (i) of the event to obtain a probability plotting position. Two of the more common formulas are Weibull... 

When a refiner changes the FCC catalyst, it is often necessary to determine the percent of the new catalyst in the unit. The following equation, which is based on a probability function, can be used to estimate the percent changeover. 

The LST alleviates this problem by systematically approximating the probabilities of Bm, with M > N, from the set of probabilities of smaller blocks, Bi, B2,. .., Bn- In this way, order correlation information is used to predict the statistical properties of evolving patterns for arbitrarily large times. The outline of the approach begins with a formal definition of block probability functions. 

Below, we will define a canonical procedure for constructing probabilities of blocks of arbitrary lengths consistent with a given block probability function P. The Kolmogorov consistency theorem will then allow us to use this set of finite block probabilities to define a measure on the set of infinite configurations, F. 

The Local Structure Operator By the Kolmogorov consistency theorem, we can use the Bayesian extension of Pn to define a measure on F. This measure -called the finite-block measure, /i f, where N denotes the order of the block probability function from which it is derived by Bayesian extension - is defined by assigning t.o each cylinder c Bj) = 5 G F cti = 6i, 0 2 = 62, , ( j — bj a value equal to the probability of its associated block ... 

The essence of the LST for one-dimensional lattices resides in the fact that an operator TtN->N+i could be constructed (equation 5.71), mapping iV-block probability functions to [N -f l)-block probabilities in a manner which satisfies the Kolmogorov consistency conditions (equation 5.68). A sequence of repeated applications of this operator allows us to define a set of Bayesian extended probability functions Pm, M > N, and thus a shift-invariant measure on the set of all one-dimensional configurations, F. Unfortunately, a simple generalization of this procedure to lattices with more than one dimension, does not, in general, produce a set of consistent block probability functions. Extensions must instead be made by using some other, approximate, method. We briefly sketch a heuristic outline of one approach below (details are worked out in [guto87b]). 

Finally, using the physical interpretation of the quantum site-state coefficients ai , we can write down an explicit form for the probability functions p . Since 0 is defined by the list n, we must simply write down the probability that a measurement of the 2r states around a given site F will yield a 2r-tuple which is an element of n. We therefore get sums of products of absolute squares, with individual list elements contributing the terms and list elements... 

Figure 10.11 shows a smooth sigmoidal threshold function that is often used in practice. It has the same form as the transition probability function used for stochastic nets ... 

Lennard-Jones, J. E., J. Chem. Phys. 20, 1024, Spatial correlation of electrons in molecules. Study of spatial probability function using the single determinant. 

In its more advanced aspects, kinetic theory is based upon a description of the gas in terms of the probability of a particle having certain values of coordinates and velocity, at a given time. Particle interactions are developed by the ordinary laws of mechanics, and the results of these are averaged over the probability distribution. The probability distribution function that is used for a given macroscopic physical situation is determined by means of an equation, the Boltzmann transport equation, which describes the space, velocity, and time changes of the distribution function in terms of collisions between particles. This equation is usually solved to give the distribution function in terms of certain macroscopic functions thus, the macroscopic conditions imposed upon the gas are taken into account in the probability function description of the microscopic situation. 

The calculated values of p are shown in Figure 2. Positive values (left of atomic number 41.7) correspond to crest superconductors, and negative to trough superconductors. Values of Tc calculated by introducing p in equation (3) (with 0 = 120°) are represented by the curve in Figure 3, which applies to the elements and the alloys of adjacent elements. The calculation for the alloys was made by use of fractions pt of M+z, Mz, M z, M+z+1, Mz+i, and M-z+i given by a probability function exp[—0.694(a — x0)2] (x = number of electrons)... 

There is no unique solution to this or most other design problems. Any design using a single tube with an i.d. of about 7.5 in or less and with a volume scaled by S will probably function from a reaction engineering viewpoint. 

In this study computational results are presented for a six-component, three-stage process of copolymerization and network formation, based on the stochastic theory of branching processes using probability generating functions and cascade substitutions (11,12). 

Thus, they share exactly the same solution (H) and performance criteria (y ) spaces. Furthermore, since their role is simply to estimate y for a given X, no search procedures S are attached to classical pattern recognition techniques. Consequently, the only element that dilfers from one classification procedure to another is the particular mapping procedure / that is used to estimate y(x) and/ or ply = j x). The available set of (x, y) data records is used to build /, either through the construction of approximations to the decision boundaries that separate zones in the decision space leading to different y values (Fig. 2a), or through the construction of approximations to the conditional probability functions, piy =j ). 

In this expression, the propagator is decomposed into the spin density at the initial position, p(q), and the conditional probability that a particle moves from q to a position q + R during the interval A, P(ri rj + R, A). Note that this function cannot be directly retrieved because it is hidden in the integral. The propagator experiment is one-dimensional however, by making use of more complex and higher-dimensional experiments, conditional probability functions such as the one mentioned here can indeed be determined (see Section 1.6 for examples). 

To compute these probabilities in Microsoft Excel, put the value of x in cell B2, say, and use the functions... 

Great simplification is achieved by introducing the hypothesis of independent reaction times (IRT) that the pairwise reaction times evolve independendy of any other reactions. While the fundamental justification of IRT may not be immediately obvious, one notices its similarity with the molecular pair model of homogeneous diffusion-mediated reactions (Noyes, 1961 Green, 1984). The usefulness of the IRT model depends on the availability of a suitable reaction probability function W(r, a t). For a pair of neutral particles undergoing fully diffusion-con-trolled reactions, Wis given by (a/r) erfc[(r - a)/2(D t)1/2] where If is the mutual diffusion coefficient and erfc is the complement of the error function. 

---