Probability of sequence

Statistical characteristics of the second type define the microstructure of copolymer chains. The best known characteristics in this category are the fractions P [/k) (probabilities) of sequences Uk involving k monomeric units. The simplest among them are the dyads U2, the complete set of which, for example, for a binary copolymer is composed of four pairs of monomeric units M2M, M2M2. The number of the types of k-ad in chains of m-component copolymers grows exponentially as mk so that with practical purposes in mind it is generally enough to restrict the consideration to sequences Uk] with moderate values of k. Their calculation turns out to be rather useful... 

The numerator in both these equations is identical. It expresses the probability of sequence A, generation in the chains. The denominators correspond to... 

The groups of Windle (1) and Blackwell (2) have demonstrated that particular sequences of monomers in thermotropic LC random copolyesters lead to very small crystalline regions in the solid state. Based on a statistical treatment of random copolymerization one concludes that the probability of sequences matching in two different chains, and hence the degree of crystallinity, increases as the chain length is lowered (1). 

Unconditional probabilities of sequences smaller than those evaluated in step three, if any, are calculated by addition of unconditional probabilities evaluated in step three. Unconditional probabilities of sequences larger than those evaluated in step three are calculated by milltipiication of appropriate conditional and unconditional probabilities. 

The entropy of copolymer 1 can be written in terms of the random sequence distribution of the copolymer. Although this is discussed in greater detail in later sections of the textbook, here the entropy of copolymerization is written in terms of the probabilities of sequences of a repeat unit, A, in the copolymer. Thus ... 

Probabilities of sequences. Assume that the four bases A, C, T, and G occur with equal likelihood in a DNA sequence of nine monomers. 

The approach based on probabilistic languages theory proposed by Garg et al. needs to give a priori the p-language (the set of sequences and for each sequence 5 the occurrence probability of sequences sub set that have like prefix). In other words, the p-language is built from equation (3). This knowledge (the whole set of sequences and their probabilities) is not aware in the dependability studies. In dependability studies, the problem is rather reverse giving a system, we need to determine the events sequences and, after that, to calculate then-occurrence probabilities. Thus, we propose to work with, as initial data, the p-automaton. 

The methodology was designed based on a traditional QRA approach. However, for applying this method in an Indian situation, several assumptions had to be made. These included the frequency of initiation events, the probability of sequence... 

In the researches were used them both for qualitative analysis to understand the situation which affected reliability problem, and for quantitative analysis to determine the probability of sequences of events. It is noted and accepted as an assumption for research, that although the observed processes and their service in each of the objects may vary, criteria relevant to the reliability of the information system are the same. From the analysis of the research presented in Figures 5-6 shows that the biggest distribution network problems are the incorrect data in the system generated by both external customers (data input to the operator system via the internet) and employees of the operator (direct manual data input to the system via computer workstation). 

We assume that the sequential errors are not correlated in time, we can write the probability of sampling a sequence of errors as the product of the individual probabilities. We further use the finite time approximation for the delta function and have ... 

The probability of B reacting is rp and the fraction of these reactions with Af molecules is rpp. The probability of the entire sequence is... 

Next let us consider the probability of finding a sequence of repeat units in a copolymer which is exactly Ui units of Mi in length. This may be represented as M2(Mi)i jM2. Working from left to right in this sequence, we note the following ... 

If the sequence contains exactly units of type Mi, then the next step must be the addition of an M2 unit. The probability of such an addition... 

Statistical considerations make it possible to test the assumption of independent additions. Let us approach this topic by considering an easier problem coin tossing. Under conditions where two events are purely random-as in tossing a fair coin-the probability of a specific sequence of outcomes is given by the product of the probabilities of the individual events. The probability of tossing a head followed by a head-indicated HH-is given by... 

The probabilities of the various dyad, triad, and other sequences that we have examined have all been described by a single probability parameter p. When we used the same kind of statistics for copolymers, we called the situation one of terminal control. We are considering similar statistics here, but the idea that the stereochemistry is controlled by the terminal unit is inappropriate. The active center of the chain end governs the chemistry of the addition, but not the stereochemistry. Neither the terminal unit nor any other repeat unit considered alone has any stereochemistry. Equations (7.62) and (7.63) merely state that an addition must be of one kind or another, but that the rates are not necessarily identical. 

Nature Consider an experiment in which each outcome is classified into one of two categories, one of which will be defined as a success and the other as a failure. Given that the probability of success p is constant from trial to trial, then the probabinty of obseivdng a specified number of successes x in n trials is defined by the binomial distribution. The sequence of outcomes is called a Bernoulli process, Nomenclature n = total number of trials X = number of successes in n trials p = probability of obseivdng a success on any one trial p = x/n, the proportion of successes in n triails Probability Law... 

A logic model that graphically portrays the range of outcomes from the combinations of events and circumstances in an accident sequence. For example, a flammable vapor release may result in a fire, an explosion, or in no consequence depending on meteorological conditions, the degree of confinement, the presence of ignition sources, etc. These trees are often shown with the probability of each outcome at each branch of the pathway... 

The likelihood function is an expression for p(a t, n, C), which is the probability of the sequence a (of length n) given a particular alignment t to a fold C. The expression for the likelihood is where most tlireading algorithms differ from one another. Since this probability can be expressed in terms of a pseudo free energy, p(a t, n, C) x exp[—/(a, t, C)], any energy function that satisfies this equation can be used in the Bayesian analysis described above. The normalization constant required is akin to a partition function, such that... 

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