Elementary Probability Theory Research Articles

A. V. Skorokhod elements of probability theory and random processes

A. V. Skorokhod elements of probability theory and random processes

Elements of probability theory, by Robert Fortet (translated from the original French). Pp xix, 524. £30. 1977. SBN 0 677 02110 0 (Gordon and Breach)

Elements of probability theory, by Robert Fortet (translated from the original French). Pp xix, 524. £30. 1977. SBN 0 677 02110 0 (Gordon and Breach)

An elementary treatment of the genetical theory of kin-selection

Hamilton's (coefficient of relation) rule for the spread of a gene influencing altruistic behavior is derived under a simple model (discrete generations, single locus, infinite population, panmictic mating) using elementary probability theory. The rule holds quite well with slow selection and it makes no difference whether the gene is dominant or recessive. The rule is altered, however, if the pool of relatives (in interaction) is finite, and I derive a correction factor for interaction between sibs in a sexual haploid. In general the ratio of benefit to cost (necessary for the altruist gene to be favored) increases as the family size decreases.

Elementary Probability Theory with Stochastic Processes

Elementary Probability Theory with Stochastic Processes

Elementary Probability Theory with Stochastic Processes.

Elementary Probability Theory with Stochastic Processes.

Elementary probability theory with stochastic processes, by Kai Lai Chung. Pp x, 325. DM 29·40. 1974. SBN 3 540 90096 9 (Springer)

Elementary probability theory with stochastic processes, by Kai Lai Chung. Pp x, 325. DM 29·40. 1974. SBN 3 540 90096 9 (Springer)

Elementary Probability Theory with Stochastic Processes.

Elementary Probability Theory with Stochastic Processes.

Bounding Distributions for Stochastic Logic Networks

A stochastic logic network is defined as a connected set of logic and time delay elements. Each of the latter elements has an associated probability distribution describing the nature of that element's delay. When used, for example, in project planning and scheduling, combinations of logic and time delay elements in such networks may represent conditions for the starting of project activities which are themselves represented by time delay elements. It is at present not known how to calculate the probability distributions for the events in such a network. This paper shows how to obtain upper and lower bounds for these probability distributions. The method is not a simulation technique; rather, it is a straightforward computational scheme derived from elementary probability theory. An example is given where the method is applied to a stochastic project scheduling network in which alternative ways exist for carrying out one of the jobs in the network.

Some Extremal Problems in Elementary Probability Theory

Some Extremal Problems in Elementary Probability Theory

Some Extremal Problems in Elementary Probability Theory

Some Extremal Problems in Elementary Probability Theory

Twin concordance: A set theoretic and probability theory approach

The concepts and assumptions used in twin concordance studies are made explicit with a mathematical treatment. The logic connecting those assumptions to the usual interpretations of twin studies is substantiated in a rigorous manner through the use of elementary set and probability theory.

Generating Functions in Elementary Probability Theory

Generating Functions in Elementary Probability Theory

Generating Functions in Elementary Probability Theory

For many years certain authors of textbooks in and statistics have used the term without first defining generating As in Wilks (1, p. 114) the usual definition of of a random variable X (with moments of all orders) is E(e8X) where stands for expected value. Also as in Wilks (1, p. 114) the same author may define probability of an integral-valued random variable X as E(sx) without letting the reader in on the fact that he is using the term generating in two different ways. Worse yet, as in Wilks (1, p. 114) the author may refer to E(sx) as the moment Although E(sx) can be and should be used to find factorial moments, it does not generate them in either of the two ways in which the previously mentioned functions generate their respective sequences. In this paper we are interested in three sequences: (1) the sequence of probabilities { pi, i = 0, 1, 2, ... where pi = P(X = i), i.e. X is an integral valued random variable; (2) the sequence of moments { E (Xi) , i = 0, 1, 2, ... I}; and (3) the sequence of factorial moments {E(X[iJ), i = 0,1,2, .... For a sequence of real numbers {ai, i = 0, 1, 2, . . .} there are two kinds of functions in common use as in Riordan (2, p. 19): the ordinary function and the exponential function. We define these functions for the given sequences.

Elements of Probability Theory. By J. Bass. Pp. ix, 249. 78s. 1966. (Academic Press.)

Elements of Probability Theory. By J. Bass. Pp. ix, 249. 78s. 1966. (Academic Press.)

Elements of Probability Theory.

Elements of Probability Theory.

THE CLINICAL APPLICATION OF BAYES' THEOREM

The doctor is ill-prepared to face up to the approaching computer revolution which will affect clinical medicine, and diagnosis especially. A machine-aided diagnosis will be expressed as the probability that a given disease is present when certain independent symptoms have been observed. This will require accurate data for the population incidence of disease and the incidence of symptoms in individual diseases. This paper gives the algebraic background, showing how Bayes' theorem is applicable to diagnosis and its potential and limitations. The derivation of Bayes' theorem, in a form suitable for coping with several symptoms and diseases, calls on the elements of probability theory, and the rules for combining probabilities in " either/or " and " and " situations.

Elements of Probability Theory.

Elements of Probability Theory.

Approximate Solutions to the Three-Machine Scheduling Problem

This paper summarizes experimental results from applying several computational methods for solving the classic three-machine scheduling model. The basic mathematical problem is to select an optimal permutation of n items, where the objective function employed is the total amount of processing time elapsing for the completion of all n items on three machines. The methods tested are integer linear programming, ordinary linear programming with answers rounded to integers, a heuristic algorithm, and random sampling. Throughout the experimentation n = 6. A dual integer programming code is tested on six trial problems. Consideration is given to studying the phenomenon of input-form sensitivity. The results are not encouraging and confirm previous experimental evidence as to the strong effect of varying the input form. However, the addition of a powerful bound on the objective function has a significant beneficial effect on the convergence of the dual algorithm. A sampling study of the statistical characteristics of this model is made with 100 sets of data randomly generated. Frequency distributions of minimum and mean total processing time are tabulated and analyzed. On this same set of problems, a linear programming approach is examined for effectiveness in producing nearly optimal schedules. Such an approach tends to give answers that are better than would be obtained by a single permutation drawn at random, but the LP rounded solutions average about an 11 per cent increase over the optimal processing time. A heuristic algorithm based on Johnson's method is tested on 20 of the 100 cases and gives excellent results. Finally attention is turned to the method of randomly sampling the permutations, possibly applying a simple heuristic gradient method to improve each sampled schedule. A feature of this method is that elementary probability theory can be applied to yield corresponding statements about the accuracy guaranteed by the approach.

Use and Abuse of Statistics

By W. J. Reichmann London: Methuen & Co. Ltd. 1963 .Pp. 336. Price 22s. 6d. The first of these books will give beginners a good start in set theory, Boolean algebra and symbolic logic, and in applications of related techniques to switching and relay circuits (including some basic computer requirements). Some elementary probability theory is also discussed in terms of set-inclusion.

The assessment of explosion hazards from electrical equipment on aircraft in the light of elementary probability theory

The paper discusses the influence of statistical considerations on the design of tests intended to lead to an assessment of the explosion risks arising from the normal operation of electrical equipment on aircraft The statistical approach implies that a finite amount of testing cannot establish a nil risk, the amount of testing depends upon the degree of assurance required. It is shown that an apparently ‘reasonable’ degree of assurance leads to an unmanageable amount of test effort if the tests are made under fully simulated service conditions.Test effort can be reduced by orders of magnitude if it is possible to assess separately each of several factors that are simultaneously necessary for an explosion Testing can be further reduced when it is possible to speed up the operation of the unit for test purposes. The proposition that test effort may be reduced by using ‘enhanced’ test conditions is considered. In this Connection, dependence on extrapolation procedures is particularly criticized.It is noted that standardization of enclosures for equipment would much reduce test effort. The advantage in relieving the test situation, of dependence on sealed enclosures that need only mechanical testing is emphasized.