Statistics continuous

N.L. Johnson and S. Kotz, Distributions in Statistics Continuous Univariate Distributions — I, Wiley, New York, 1970. 

The concept of a fluidized bed consisting of electrically conducting particles as a statistically continuous electrode was first discussed by Le Goff et al. (Lie). Interesting similarities with heat-transfer studies in fluidized beds may be exploited to advantage by use of the limiting current method. 

Johnson, N.L. and Kotz, S. (1970). Distributions in Statistics. Continuous Univariate Distributions - I. John Wiley Sons, Ltd, New York. 

Statistics Continuous variable Discrete variable Sample... 

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

The set of microstates of a finite system in quantum statistical mechanics is a finite, discrete denumerable set of quantum states each characterized by an appropriate collection of quantum numbers. In classical statistical mechanics, the set of microstates fonn a continuous (and therefore infinite) set of points in f space (also called phase space). 

The principal tool for performance-based quality assessment is the control chart. In a control chart the results from the analysis of quality assessment samples are plotted in the order in which they are collected, providing a continuous record of the statistical state of the analytical system. Quality assessment data collected over time can be summarized by a mean value and a standard deviation. The fundamental assumption behind the use of a control chart is that quality assessment data will show only random variations around the mean value when the analytical system is in statistical control. When an analytical system moves out of statistical control, the quality assessment data is influenced by additional sources of error, increasing the standard deviation or changing the mean value. 

The statistical nature of polymers and polymerization reactions has been illustrated at many points throughout this volume. It continues to be important in the discussion of stereoregularity. Thus it is generally more accurate to describe a polymer as, say, predominately isotactic rather than perfectly isotactic. More quantitatively, we need to be able to describe a polymer in terms of the percentages of isotactic, syndiotactic, and atactic sequences. 

When we discussed random walk statistics in Chap. 1, we used n to represent the number of steps in the process and then identified this quantity as the number of repeat units in the polymer chain. We continue to reserve n as the symbol for the degree of polymerization, so the number of diffusion steps is represented by V in this section. 

More recently, studies of wine and beer have initiated techniques of statistically vaUd sensory analysis. Scientific studies involving wine continue in these areas, building on past discoveries. Natural phenols as desirable dietary components and monitors of storage and aging reactions are currently active fields. Viticultural research, as well as enological, continues to improve grapes and the wines made from them (11). 

Manufacturing processes have been improved by use of on-line computer control and statistical process control leading to more uniform final products. Production methods now include inverse (water-in-oil) suspension polymerization, inverse emulsion polymerization, and continuous aqueous solution polymerization on moving belts. Conventional azo, peroxy, redox, and gamma-ray initiators are used in batch and continuous processes. Recent patents describe processes for preparing transparent and stable microlatexes by inverse microemulsion polymerization. New methods have also been described for reducing residual acrylamide monomer in finished products. 

Clinical trials for r-IEN-y in RA indicated that the dmg is well tolerated (52). Consistent improvement in tender and swollen joint scores was observed, but a large number of patients were needed in the trial to show statistical significance for r-IEN-y treatment. In certain individuals, responses were remarkable. An additive effect between r-IEN-y and penicillamine was detected. Efficacy was lower when r-IEN-y was combined with gold therapy. Research is continuing. 

Diffusion of Carbon. When carbon atoms are deposited on the surface of the austenite, these atoms locate in the interstices between the iron atoms. As a result of natural vibrations the carbon atoms rapidly move from one site to another, statistically moving away from the surface. Carbon atoms continue to be deposited on the surface, so that a carbon gradient builds up, as shown schematically in Figure 5. When the carbon content of the surface attains the equihbrium value, this value is maintained at the surface if the kinetics of the gas reactions are sufficient to produce carbon atoms at least as fast as the atoms diffuse away from the surface into the interior of the sample. 

Reliability. There has been a significant rise in interest among pump users in the 1990s to improve equipment reflabiUty and increase mean time between failures. Quantifiable solutions to such problems are being sought (61). Statistical databases (qv) have grown, improved by continuous contributions of both pump manufacturers and users. Users have also learned to compile and interpret these data. Moreover, sophisticated instmmentation has become available. Examples are vibration analysis and pump diagnostics. 

Quality in Japan. Japanese economic prowess has been attributed variously to such quahty improvement activities as quahty circles, statistical process control (SPG), just-in-time dehvery (JIT), and zero defects (ZD). However, the real key to success hes in the apphcation of numerous quahty improvement tools as part of a management philosophy called Kaizen, which means continuous improvement (10). 

Gaussian plume models for continuous sources can be obtained from statistical arguments or can be derived by solving ... 

Statistical quality control is used to first measure and then continuously improve product quality. For example, The Dow Chemical Company s average 1989 performance compared to the typical sales specification were purity, = 99.65 wt % color, APHA = 4 acid (as HCl) = 7.3 ppm and water = 26 ppm. Averages of properties were based on rail car and tank tmck shipment samples during 1989. 

For certain types of stochastic or random-variable problems, the sequence of events may be of particular importance. Statistical information about expected values or moments obtained from plant experimental data alone may not be sufficient to describe the process completely. In these cases, computet simulations with known statistical iaputs may be the only satisfactory way of providing the necessary information. These problems ate more likely to arise with discrete manufactuting systems or solids-handling systems rather than the continuous fluid-flow systems usually encountered ia chemical engineering studies. However, there ate numerous situations for such stochastic events or data ia process iadustries (7—10). 

Nicotrol, developed by Cygaus Corp. and to be marketed by Warner-Lambert, is the fourth nicotine transdermal system to be approved ia the Uoited States. It dehvers nicotine only duriag wakiag hours to mimic better the habits of smokers (109). la oae coatroUed study, ao statistical differeaces ia efficacy and good tolerabihty were observed for the wakiag and continuous regimen (112). 

For example, the measured pressure exerted by an enclosed gas can be thought of as a time-averaged manifestation of the individual molecules random motions. When one considers an individual molecule, however, statistical thermodynamics would propose its random motion or pressure could be quite different from that measured by even the most sensitive gauge which acts to average a distribution of individual molecule pressures. The particulate nature of matter is fundamental to statistical thermodynamics as opposed to classical thermodynamics, which assumes matter is continuous. Further, these elementary particles and their complex substmctures exhibit wave properties even though intra- and interparticle energy transfers are quantized, ie, not continuous. Statistical thermodynamics holds that the impression of continuity of properties, and even the soHdity of matter is an effect of scale. 

There are important economic consequences of a process being out of control for example, product waste and customer dissatisfaction. Hence, statistical process control does provide a way to continuously monitor process performance and improve product quahty. A typic process may go out of control due to several reasons, including... 

In continuous processes where automatic feedback control has been implemented, the feedback mechanism theoretically ensures that product quality is at or near the set point regardless of process disturbances. This, of course, requires that an appropriate manipulated variable has been identified for adjusting tne product quality. However, even under feedback control, there may be daily variations of product quahty because of disturbances or equipment or instrument malfunctions. These occurrences can be analyzed using the concepts of statistical quahty control. 

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