Probability random

Papoulis, A., Probability, Random Variables, and Stochastic Processes, McGraw-Hill Singapore, 1991... 

Based on experiments with linearly polarized light, Mayer also concluded that the photoreceptor is arranged in a dichroic fashion close to the cell wall. The electrical dipole moment for the absorption of bluelight lies parallel to the cell wall, but is probably random with respect to the normal of the cell wall. In the first experiment, the cells were irradiated with bright light. Clearly, the chloroplasts separate from the walls, which are parallel to the -vector and exhibit a banded pattern (Fig. 17, left). However, in weak polarized light the chloroplasts tended to move close to those walls parallel to the -vector (Fig. 17, right). 

H. Stark and J. W. Woods, Probability, Random Processes and Estimation Theory for Engineers, Prentice Hall, Englewood Cliffs, 1986. 

Papoulis, A., 1965, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York. Pauling, L., 1935, The oxygen equilibrium of hemoglobin and its structural interpretation, Proc. Natl. 

Quantitative measurements of the lower MW band revealed the presence of several Si-Hs in each PDMS chain. Furthermore, the shapes of the DR and IR traces are the same, which indicates that the distribution of hydride groups is proportional to MW and probably random. An example of a non-random distribution would be a PDMS with Si-H terminated chains. With two hydrides per chain, regardless of size, the IR trace of the lower MW portion would be enhanced and the shapes of the two curves would differ markedly. 

Davis, M.C., IEEE Trans, on Automatic Control, 1963, AC-8, I96. 2h. Papoulis, A., "Probability, Random Variables and Stochastic Processes", McGraw-Hill 1965. 

If an alkyl- or aryltrichlorosilane is treated, in bulk or in solution, with a considerable excess of water, an amorphous, infusible, insoluble product is usually formed with the approximate empirical composition, (RSi01-5)x or (ArSiOi.g). Insolubility has precluded estimation of the molecular weights of these products, and the amorphous appearance has discouraged crystallographic studies. These amorphous polymers are probably randomly cross-linked. Only in recent years have definable, low polymeric components of this composition been isolated and identified. 

A. Papoulis, Probability, Random Variables and Stochastic Processes, New York McGraw-Hill, 1965. 

The action of B. subtilis alpha-amylase is probably random on the internal bonds of amylose, but non-random attack occurs near chain ends. Further calculations indicate that little, if any, multiple attack occurs. Few results have been obtained for the enzyme from A. oryzae, but it is believed that multiple attack does not occur. ... 

The observed frequencies with 3, 5, 7 and 14 different and equally probable random error sources are shown in Fig. 3,6c-f. With many different sources of experimental error, it is seen that the frequency of the experimental response data can be approximately described by the bell-shaped curve in Fig. 3.6g. 

The probable random error in the mean of a set of repeated measurements can be determined statistically. 

The assumption that the contraction process is ideally adiabatic, while perhaps not entirely permissible practically, seems indicated by modern theory of the behavior of molecular chains, which pictures these as undergoing, when freed of restraints, a sort of segmental diffusion, much like the adiabatic expansion of an ideal gas into a vacuum (155). In the case of the molecular chain, it diffuses to the most probable, randomly coiled configuration, which is much less asymmetric, hence shorter, than an initially extended chain. Because rubber most nearly presents this ideal behavior, those fibers which develop increased tension (a measure of the tendency toward assumption of the contracted form) when held isometrically under conditions of increasing temperature (favoring the diffusion ) are said to be rubber-like. Most normal elastic solids upon stress are strained from some stable structure and relax as the temperature is raised. 

This chapter reviews basic ideas of probability, random variables, probability distributions, and sampling distributions of a statistic. The axiomatic approach is followed to define probabilities of events in a sample space, which is the collection of all possible outcomes of an experiment. When the outcomes are equally likely, the probability of an event is just the ratio of the number of outcomes in the event to the number of outcomes in the sample space. 

Annual operating savings (a) Probability (b) Corresponding random numbers (c) Project life (d) Couesponding Probability random numbers (e) (f) ... 

Let us assume they represent N = 7 independent measurements (this is often not the case, since values from the same source are quoted in several references). Then the explosion limit may be assumed to be a random variable, i.e. a variable which adopts certain values with a certain probability. Random variables are described by probability distributions (vid. Appendix C). In what follows the logarithmic normal (lognormal) distribution (vid. Sect. 9.3.4) is used to represent the values... 

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