Probability functions

We have already found that the probability function governing observation of a single event x from among a continuous random distribution of possible events x having a population mean p and a population standard deviation a is... 

Fig. 1.14 Normal and skew probability functions. (Courtesy DallaValle )...

Well-behaved probability functions total unity when they are summed over all possible outcomes. Since Eq. (1.31) is a continuous function-this has been accomplished by getting rid of the factorials-this sum may be written as an integral over all possible values of x ... 

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... 

The overall probability function for the end-to-end distance is the product of these two considerations. Starting at r = 0, the probability increases owing to the r term, passes through a maximum, then decreases as the exponential factor takes over at large r values. 

Now we recognize that the weighting factor fj is precisely what the probability function gives, so we write Eq. (1.45) as... 

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... 

Converting the probability function described above into an excluded volume is accomplished by integrating the probability 1 - 0(d) of exclusion over a spherical volume encompassing ah values of d ... 

The probability function that has been displayed is a very special case of the more general case, which is called the binomial probabihty distribution. 

PCu(ci,q) is clearly not a 5-function as has been suggested. Many more LSMS calculations would have to be done in order to determine the structure of Pcn(ci,q) for fee alloys in detail, but it is easier to see the structure in the conditional probability for bcc alloys. The probability Pcu(q) for finding a charge between q and q-t-dq on a Cu site in a bcc Cu-Zn alloy and three conditional probabilities Pcu(ci,q) are shown in Fig. 6. These functions were obtained, as for the fee case, by averaging the LSMS data for the bcc alloys with five concentrations. The probability function is not a uniform function of q, but the structure is not as clear-cut as for the fee case. The conditional probabilities Pcu(ci,q) are non-zero over a wider range than they are for the fee alloys, and it can be seen clearly that they have fine structure as well. Presumably, each Pcu(ci,q) can be expressed as a sum of probabilities with two conditions Pcu(ci,C2,q), but there is no reason to expect even those probabilities to be 5-functions. 

Although we work with the pair distribution functions, what we are to solve are essentially the point probability functions fj(rj), i=A and B. 

For the PPM, corresponding to the free energy of the CVM is the Path Probability Function (hereafter PPF), P t t -t- At), which is an explicit function of time and is defined as the product of three factors Pj, P2 and P3. Each factor is provided in the following in the logarithmic expression. 

A bounded continuous random variable with uniform distribution has the probability function... 

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

The probability function for the standard normal distribution is then... 

The chi-square distribution gives the probability for a continuous random variable bounded on the left tail. The probability function has a shape parameter... 

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. 

Block Probability Functions We define the block probability function of order-N, Pn, to be a map from Bat, Bat-i,. .., Bq into the reals that satisfies the... 

The set of all block probability functions of order-V is denoted by Pat. If two block probability functions, and Pm, with M > N, are self-consistent and are equal on blocks of size. s < N, they are said to be mutually consistent. 

The Kolmogorov consistency theorem [gnto88] asserts that any set of self- and mutually- consistent probability functions Pj, j = 1,2,..., jV may be extended to a unique shift-invariant measure on F,... 

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. 

It is an easy exercise to show that if Pn satisfies the Kolmogorov consistency conditions (equations 5.68) for all blocks Bj of size j < N, then T[N- N+LPN) satisfies the Kolmogorov consistency conditions for blocks Bj of size j < N + 1. Given a block probability function P, therefore, we can generate a set of block probability functions Pj for arbitrary j > N hy successive applications of the operator TTN-tN+i, this set is called the Bayesian extension of Pn-... 

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 ... 

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