Probability distributions characteristic parameters

By including characteristic atomic properties, A. of atoms i andj, the RDF code can be used in different tasks to fit the requirements of the information to be represented. The exponential term contains the distance r j between the atoms i andj and the smoothing parameter fl, which defines the probability distribution of the individual distances. The function g(r) was calculated at a number of discrete points with defined intervals. 

The sample of individuals is assumed to represent the patient population at large, sharing the same pathophysiological and pharmacokinetic-dynamic parameter distributions. The individual parameter 0 is assumed to arise from some multivariate probability distribution 0 / (T), where jk is the vector of so-called hyperparameters or population characteristics. In the mixed-effects formulation, the collection of jk is composed of population typical values (generally the mean vector) and of population variability values (generally the variance-covariance matrix). Mean and variance characterize the location and dispersion of the probability distribution of 0 in statistical terms. 

In the mixed-effects context, the collection of population parameters is composed of a population-typical value (generally the mean) and of a population-variability value (generally the variance-covariance matrix). The mean and variance are the first two moments of a probability distribution. They build a minimal set of hyperparameters or population characteristics for it, which is sufficient (in a statistical sense) when F is taken as normal or log-normal. 

ReliahUity measurement tests are used to make estimates of the reliahUity of a system or a population of items. Parametric and nonparametric estimates are used. Parametric estimates are based on a known or assumed distribution of the system characteristic of interest. The parameters are the constants that describe the shape of the distribution. Nonparametric estimates are used without assuming the nature of the underlying probability distribution. Generally, nonparametric estimates are not as efficient as parametric estimates. Nonparametric reliability estimates apply only to a specific test interval and cannot be extrapolated. Parametric estimates are described in this section when the underlying distribution is exponential and WeibuU. The three types of parametric estimates that are frequently used... 

In (Burgazzi 2003, Burgazzi 2007) the quantification of the probability of failure of the system is accom-phshed assuming a fixed and unchanging distribution (with constant mean and standard deviation values over a time interval) for both the threshold and actual values of the characteristic parameter, thus implying the time independence of the characteristic parameter. As stressed before, here we treat the parameters as time dependent upon time, starting from expression (1), which defines the time dependent limit state function. 

Another random parameter of importance in practical situations is the emission time of the photon wave packets from the source, which is described by a probability distribution with a time width AT characteristic of the macroscopic time-resolved operation of the source. The total time duration of the pulses is hence... 

It is important to distinguish between parameters of statistical distributions such as density function, expected value, and standard deviation on the one hand, and corresponding sample characteristics such as histograms, mean value and empirical standard deviation on the other. The latter are calculated from actual samples and will converge — in a stochastic sense — with increasing sample size towards the distribution parameters which are the usually unknown characteristics of the abstract probability distribution behind infinitely many samples of one and the same experiment. The sample-based quantities are also called estimates of the corresponding statistical parameters. 

In the current work, we are not concerned with providing improved statistical models. Rather, we will use already existing models as given in the literature. By applying either Eq. (2.12) for a conventional RIS model or Eq. (2.14) for advanced Monte Carlo based models, we can infer the probability distribution p4(A>...>t 4) which will be important for the simulation of the NMR spectra. On the other hand, global parameters (characteristic ratio, etc.) [5] are frequently calculated from RIS models for comparison with experiment. Because of this interplay of local and global geometry and to avoid a clumsy notation, RIS model and random coil picture are used quite indiscriminately in the current work. 

Another exceedingly important validation step for emphatically flexible molecules is to test whether steric parameters derived by averaging the characteristics of a diverse conformational set are capable of reproducing the flexibility of species from the smdied series at the MD level. Due to the unusual behavior of the torsion angles in this class of compounds (the pronounced gauche-effect), it is necessary to generate the probability distribution of the parametrized dihedral angle C-O-C-C and the symmetric C-C-O-C in the MD simulation of diethyl ether in liquid phase in order to check the performance of the modified force field. 

For each probability distribution and each bumup, 3,000 groups of random system parameters and factors are sampled. Altogether, 18,000 groups of data are calculated. MCST is calculated for each group of the samples. The calculated MCST distributions of case 1 and case 2 for BOC, MOC and EOC are represented in Fig. 2.92 [27]. The evaluated statistical characteristics of the MCST distributions are summarized in Table 2.17 [27]. 

For smectic phases the defining characteristic is their layer structure with its one dimensional translational order parallel to the layer normal. At the single molecule level this order is completely defined by the singlet translational distribution function, p(z), which gives the probability of finding a molecule with its centre of mass at a distance, z, from the centre of one of the layers irrespective of its orientation [19]. Just as we have seen for the orientational order it is more convenient to characterise the translational order in terms of translational order parameters t which are the averages of the Chebychev polynomials, T (cos 2nzld)-, for example... 

In order to define the statistical characteristics of a many particle system, for instance an ideal gas, their distribution function with some defined physical parameters (for example, velocity, momentum, energy, etc) should be fully determined. In particular it is physically important to define the velocity of particles corresponding to the most probable state, which is the maximum of the distribution function. 

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