Chebyshev's Theorem Research Articles

Concerning the motion of the center of mass of a crank mechanism

After explaining the general background, it is shown by the author how one may generate, as a coupler curve the trajectory, of the total center of mass of the movable links of a four-bar linkage. This is done by means of basic configurations, which have been supplemented by appropriate parallelogram mechanisms, and by reference to the Roberts—Chebyshev theorem. The work also gives directions for the construction of a crank mechanism (four-bar linkage) whose center of mass trajectory has been generated as a coupler curve.

Credibility Theory: A Probabilistic Development

Credibility theory is the term used by casualty actuaries to encompass procedures for weighting and combining data from different sources into parameter estimates upon which insurance rates are based. Criticism has been leveled at these procedures, frequently as a result of a lack of understanding of the principles underlying their use. This article explains the theory of credibility procedures by developing the concept from such basic theorems of probability as Chebyshev's Theorem, the Central Limit Theorem, and the Law of Large Numbers. Credibility theory is the term used by casualty actuaries to encompass those procedures used for weighting and combining data from different sources into a parameter estimate upon which some insurance rates are based. When defined in this manner, these techniques or procedures and the problems associated with them are neither new nor are they unique to the field of insurance. Nevertheless these procedures have frequently been criticized within the insurance industry either on the grounds that the nature of the problem is such that the best basis for an insurance rate is the informed judgment of an experienced casualty underwriter or on the grounds that there appears to be insufficient mathematical justification for the use of these credibility practices. The determination of whether or not these criticisms are justified would appear to be a worthwhile undertaking. If one can enlarge the definition of the word to include not only the Jerome D. Braverman, Ph.D., is Associate Professor in the Rochester Institute of Technology. Previously he was Senior Staff Engineer at Space Systems Division of Hughes Aircraft Company and Lecturer in Statistics at San Fernando Valley State College. This paper was submitted in June, 1967. physical origin of a set of data, but also the temporal origin as well, one draws closer to the heart of the problem. For, in the process of insurance rate-making, the actuary is frequently faced with data derived from or collected during different time periods. His problem then is not basically different from that of the economic forecaster or the industrial statistician testing or evaluating the parameter of some continuing process. The question of primary interest concerns an explanation of the deviations from some previously accepted value of that parameter. Do these deviations result from a change in the underlying process or are they merely the result of purely random fluctuations in the data? This specific problem is part of a larger problem of parameter estimation and the relationship of prior information, regardless of its source or derivation, to sample or experimental data. Arthur L. Bailey, in 1950, remarked that while casualty underwriters believe that they are not entirely devoid of knowledge before they have acquired any statistical data, standard statistical works make the tacit assumption that no knowledge existed prior

On the greatest prime factor of a quadratic polynomial

as x ~ ~ , for which as for many other interesting results in the theory of numbers we are indebted to Chebyshev, has a t t racted the interest of several mathematicians. Revealed posthumously as little more than a fragment in one of Chebyshev's manuscripts, the theorem was first published and fully proved in a memoir by Markov in 1895 [6], while later in the same year a generalisation by Ivanov [4] appeared in which the polynomial n2§ 1 was replaced by n2+A for any positive A (an account of both Markov's and Ivanov ' s work is to be found in Paragraphs 147 and 149 of Landau's Primzahlen [5]). In 1921 Nagell [7] improved and further generalised Chebyshev's theorem by shewing that for any e log~ x, X