Basic probability theorems

In the language of probability, an event is an outcome of one or more experiments or trials and is defined by the experimenter. Some examples of events are 

Tossing a coin twice and getting heads both times 

Tossing a coin 10 times and getting heads for the first five times and tails for the other five 

Picking up one card from a deck of cards and that card being red 

Picking up 10 cards from a deck and all of them being hearts 

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However, for some pairs of binary strings, the ordering between their occurrence probabilities is independent of the basic probabilities p and it only depends on the relative positions of their Os and Is. More precisely, the following theorem (2 3) provides us with an intrinsic order criterion—denoted from now on by the acronym IOC—to compare the occurrence probabilities of two given n-tuples of Os and Is without computing them. 

The matrix condition IOC, stated by Theorem 2.2 or by Remark 2.6, is called the intrinsic order criterion, because it is independent of the basic probabilities p, and it only depends on the relative positions of the Os and Is in the binary n-tuples u, v. Theorem 2.2 naturally leads to the following partial order relation on the set 0,1 " (3). The so-called intrinsic order will be denoted by , and we shall write u > V (u < v)to indicate that u is intrinsically greater (less) than or equal to v. The partially ordered set (from now on, poset, for short) ( 0,1 ", on n Boolean variables will be denoted by / . 

The evaluation of elements such as the M n,fin s is a very difficult task, which is performed with different levels of accuracy. It is sufficient here to mention again the so called sudden approximation (to some extent similar to the Koopmans theorem assumption we have discussed for binding energies). The basic idea of this approximation is that the photoemission of one-electron is so sudden with respect to relaxation times of the passive electron probability distribution as to be considered instantaneous. It is worth noting that this approximation stresses the one-electron character of the photoemission event (as in Koopmans theorem assumption). 

PROBABILITY An Introduction, Samuel Goldberg. Excellent basic text covers set theory, probability theory for finite sample spaces, binomial theorem, much more. 360 problems. Bibliographies. 322pp. 5H 8U. 65252-1 Pa. 7.95... 

Risk is defined as tlie product of two factors (1) the probability of an undesirable event and (2) tlie measured consequences of the undesirable event. Measured consequences may be stated in terms of financial loss, injuries, deatlis, or oilier variables. Failure represents an inability to perform some required function. Reliability is the pr( >ability that a system or one of its components will perform its intended function mider certain conditions for a specified period. The reliability of a system and its probability of failure are complementary in the sense that the sum of these two protobilities is unity. Tliis cluipter considers basic concepts and theorems of probability that find application in the estimation of risk and reliability. 

Tliis cliapter is concerned with special probability distributions and teclutiques used in calculations of reliability and risk. Theorems and basic concepts of probability presented in Cliapter 19 are applied to tlie determination of tlie reliability of complex systems in terms of tlie reliabilities of their components. Tlie relationship between reliability and failure rate is explored in detail. Special probability distributions for failure time are discussed. The chapter concludes with a consideration of fault tree analysis and event tree analysis, two special teclmiques tliat figure prominently in hazard analysis and tlie evaluation of risk. 

To determine the standard deviation of the mean lead concentration in the receiving tank, a basic theorem in probability called the central limit theorem must be employed. If a represents the standard deviation of the lead concentrations in the drums, then the standard deviation of the mean lead concentration in the receiving tank is given by a/y/n where n is the number of drums ... 

It is true that Condorcet s view is an ancient one - clearly defended, for instance, throughout Plato s Dialogues. He was what is these days called a moral realist , who assumed that there are objective moral truths and he further believed that, as people become more enlightened, the probability will increase that what most think is the right thing to do will indeed be the right thing to do (an assumption basic for his famed jury theorem ). Thus, he asks in the Esquisse ... 

One of the fundamental results in probability is the Central Limit Theorem which concerns an infinite collection of mutually independent, identically distributed random variables. The basic observation is that if we have some random variables drawn from some particular distribution we may define a new random variable by averaging. The average will not have the same distribution as those of the collection, but it will have a well defined density. The Central Limit Theorem effectively describes the limiting density obtained by averaging over an infinite collection. Remarkably, independent of the density of a particular member of the collection, this averaged random variable will ultimately be Gaussian with mean and variance specified by the mean and variance associated to the members of the collection. 

Thus, the periodicity of the density of probabilities (3.80) was lowered at the level of eigen-function such as Eq. (3.88), regaining the celebrated Bloch theorem of Eq. (3.34), here in a generalized form the eigen-function of an electron in a periodic potential can be written as a product of a function carrying the potential periodicity and a basic exponential factor exp(/lx) . 

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