Probability theory expectation

By appropriate manipulation, it is possible to determine the expected value of various functions of X, which is the subject of probability theory. For example, the expected value of X is simply the sum of squares of the values, each weighted by the probability of obtaining the value. 

In the Poisson case, the decoherence theory affords a more satisfactory justification for the correspondence principle [20]. Adopting the Wigner formalism, it is possible to express quantum mechanical problems in terms of the classical phase space, and the Wigner quasi-probability is expected to remain positive definite until the instant at which a quantum transition occurs, according to the estimate of Ref. 120, at the time... 

Theoretical chemistry has two problems that remain unsolved in terms of fundamental quantum theory the physics of chemical interaction and the theoretical basis of molecular structure. The two problems are related but commonly approached from different points of view. The molecular-structure problem has been analyzed particularly well and eloquent arguments have been advanced to show that the classical idea of molecular shape cannot be accommodated within the Hilbert-space formulation of quantum theory [161, 2, 162, 163]. As a corollary it follows that the idea of a chemical bond, with its intimate link to molecular structure, is likewise unidentified within the quantum context [164]. In essence, the problem concerns the classical features of a rigid three-dimensional arrangement of atomic nuclei in a molecule. There is no obvious way to reconcile such a classical shape with the probability densities expected to emerge from the solution of a molecular Hamiltonian problem. The complete molecular eigenstate is spherically symmetrical [165] and resists reduction to lower symmetry, even in the presence of a radiation field. 

Probability theory deals with the expected frequencies of various events in random sampling. The set of events considered in the sampling is called the sample space of the given problem, and may be discrete (like "heads" and "tails" in coin tossing) or continuous (like the set of values on the real number line). 

Upon this bedrock idea, a highly complex and sophisticated statistics has been constructed that can be used to compute not only the frequency of occurrence of most events, but also many kinds of averages such as the expectation values of ensembles of particles. Does this fundamental idea provide a sound basis for all these computations The philosopher Ayer insists that probability theory cannot yield any certainty about future events and that it cannot even indicate what is likely to happen, yet, bearing this in mind, he goes on to state his personal belief that the future will probably resemble the past and so render probabilistic interpretations of our world valid. His presumption would appear to be borne out in practice, especially where large numbers of events or large populations of entities are concerned. 

In assessing precision, it is important to be able to decide whether two bonds that differ in length are really different, or whether the observed difference is merely that expected from the imprecision of the measurements (indicated by the e.s.d. of each atomic position). The significance of diflferences in bond lengths can be approached by probability theory. The probability that x does not differ from its mean value by more than qa = p is given for various values of q, in Table 11.5. For example, if two bond lengths differ by 0.04 A, and the positional e.s.d. = 0.005 A for both atoms, the e.s.d. of a bond is ... 

There is little question that the double-commutator expressions in (3.46) and (3.47) greatly simplify the algebraic and computational aspects of the calculation of the m2 matrix elements occurring in the equation system (3.42) see Ref. 9. The theoretical results as to excitations, ionizations, etc., of certain many-particle systems are in such good agreement with experimental experience that one can probably only expect that part of this agreement will be lost, if one tries to refine the theory. 

Observables calculated from approximate wavefunctions as in Equation 1.13 are called expectation values, an expression used in probability theory. In practice, we will always have to be satisfied with approximate wavefunctions. How can we choose between different approximations And if our trial wavefunction has adjustable parameters (such as the coefficients of atomic orbitals in molecular orbitals see Section 4.1), how can we choose the adjustable parameters best values Here, Rayleigh s variation theorem is of great value. It tells us that the expectation value for the ground state energy E, (E ), calculated from an approximate wavefunction (P is always larger than the true energy E (Equation 1.14). Proof of the variation theorem is given in textbooks on quantum mechanics.18... 

From elementary probability theory, rettrm is maximized by selecting the alternative with the greatest expected value. The expected valtre of an action Aj is calculated by weighting its consequences Cj over aU events k, by the probability the event will occur. The expected value of a given action Aj... 

Principles that depend on determination of expected values by the mathematics of probability theory are frequently criticized on the grounds that the theory holds only when trials are repeated many times. It is argued that, for certain types of decisions—for example, whether to finance a major expansion—expectation is meaningless since this type of decision is not made very often. According to the counterargument, even if the firm is not faced with a large number of repetitive decisions, it should apply the principle to many different decisions and thus realize the long-run effects. Moreover, even if the decision is unique, the only way to approach decisions for which probabilities are known is to behave as if the decision were a repetitive one and thus minimize expected cost or maximize expected revenue or profit. 

On the contrary, the probability of the top event, P A), describes the analyst s (an analyst group s) uncertainty about the occurrence of the top event, reflecting both the underlying randomness as well as the lack of knowledge involved. Although based on and equal to the expected value of a limiting relative frequency, it follows from probability theory that this number encapsulates all the underlying randomness and uncertainties, and the interpretation is the one with reference to a standard. 

Through the use of basic probability theory and statistical analysis, the system safety function can actually assign expected values to certain hazards and/or failures to determine the likelihood of their occurrence. The availability of such quantifiable information further enhances the management decisionmaking process and justifies the existence of the system safety effort within the organization. 

On the fault tree, when probability rates are known, the analyst simply adds the probability values for the events under an or gate and arrives at the expected probability for the occurrence of the main or top event. However, it is noted that in many systems, events are not mutually exclusive in that more than one can occur at the same time and result in the same outcome. In fact, the OR gate simply indicates that more than one or only one event must occur to affect the main event When there is some probability that more than one event can occur at the same time to affect an outcome, it is known as joint probability. Figure 12.6 shows the overlapping effect, which creates the joint probability theory. The event labeled AB in this diagram indicates that these events would actually be counted twice should formula (12.1) be used by accidentally labeling the event as mutually exclusive. In order to... 

Uncertainty theory is also referred to as probability theory, credibility theory, or reliability theory and includes fuzzy random theory, random fuzzy theory, double stochastic theory, double fiizzy theory, the dual rough theory, fiizzy rough theory, random rough theory, and rough stochastic theory. This section focuses on the probability theory and fiizzy set theory, including probability spaces, random variables, probability spaces, credibility measurement, fuzzy variable and its expected value operator, and so on. 

The entropy Sipii,..., pij,... is a function of a set of probabilities. The distribution of p,j s that cause 5 to be maximal is the distribution that most fairly apportions the constrained scores between the individual outcomes. That is, the probability distribution is flat if there are no constraints, and follows the multiplication rule of probability theory if there are independent constraints. If there is a constraint, such as the average score on die rolls, and if it is not equal to the value expected from a uniform distribution, then maximum entropy predicts an exponential distribution of the probabilities. In Chapter 10, this exponential function will define the Boltzmann distribution law. With this law you can predict thermodynamic and physical properties of atoms and molecules, and their averages and fluctuations. How-ever, first we need the machinery of thermodynamics, the subject of the next three chapters. 

Do we expect this model to be accurate for a dynamics dictated by Tsallis statistics A jump diffusion process that randomly samples the equilibrium canonical Tsallis distribution has been shown to lead to anomalous diffusion and Levy flights in the 5/3 < q < 3 regime. [3] Due to the delocalized nature of the equilibrium distributions, we might find that the microstates of our master equation are not well defined. Even at low temperatures, it may be difficult to identify distinct microstates of the system. The same delocalization can lead to large transition probabilities for states that are not adjacent ill configuration space. This would be a violation of the assumptions of the transition state theory - that once the system crosses the transition state from the reactant microstate it will be deactivated and equilibrated in the product state. Concerted transitions between spatially far-separated states may be common. This would lead to a highly connected master equation where each state is connected to a significant fraction of all other microstates of the system. [9, 10]... 

This result comes from the idea of a variational rate theory for a diffusive dynamics. If the dynamics of the reactive system is overdamped and the effective friction is spatially isotropic, the time required to pass from the reactant to the product state is expected to be proportional to the integral over the path of the inverse Boltzmann probability. 

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