Discrete distribution, definition

One is familiar with the idea of discrete and definite electronic stales in molecules, as revealed by molecular spectroscopy. Each electronic stale possesses a number of vibrational states that are occupied to a great extent near the ground state at normal temperatures. Each vibrational state has, if the stcric conditions are enabling, a number of rotational states associated with it, and for gas molecules both the vibrational and the rotational states can easily be observed and measured spectroscopically. Correspondingly, the distribution of the vibrational states in solids (phonon spectra) is easily measurable. 

Drawing a playing card at random from a deck is another example of an experiment with an underlying discrete distribution, with equal probability (1/52) assigned to each card. For a discrete distribution, the definition of the expected value is ... 

Since the molecular weight is a distributed quantity, the concepts and properties of statistical distributions can be applied to the MWD. A statistical definition that is particularly useful is that of moment of a distribution. In statistics, the S th moment of the discrete distribution / of a discrete random variable y, is defined as... 

Another method to access aeidie groups on activated carbons is direct potentio-metric titration, but it suffers from very slow exchange equilibria in solution [113] and the lack of definite endpoints. The latter faet reflects the spread of dissociation constants of most surface funetionalities on carbons caused by interactions between neighboring groups and effeets resulting from the limited size of the graphene sheets. Assuming a continuous, rather than discrete, distribution of pK values, Contescu et al. [114] showed that potentiometric titration data could be transformed to proton affinity distribution (PAD) eurves and that the results were in... 

The intramolecular excitation of a sequence of definite electronic levels in the positive ion is the most probable explanation of the appearance of slow electrons. The differences between the electron kinetic energies for the subsequent maxima in the distribution curves are a measure of the discrete values of the excitation energy absorbed by the molecule in the process of photoionization. Table III gives a summary of these values. 

It is common practice to make a distinction between the inorganic constituents of so-called "Eastern" and "Western" coals By definition. Western coals are those for which the CaO+MgO content exceeds the Fe203 content of the ash, while the reverse is true for Eastern coals [ 1 I The inorganic constituents in Eastern coals, which are principally bituminous in rank, are predominantly in the form of discrete mineral particles. Clay minerals (kaolinite, illite) are usually dominant, followed by quartz and pyrite. The range and typical values of the mineral distribution and ash chemistry of Eastern coals are shown in Table I. These data were determined from computer-controlled scanning electron microscopy (CCSEM), Mossbauer spectroscopy, and other measurements on over a hundred coals. 

The radial distribution function, g(r), is the ratio of the density distribution of a type of site / at a given distance from a given type of site i, to the average density of site / in the system. A more technical definition, which allows the binning of this function into discrete intervals of r, is ... 

The first uses a model of the scattering system to evaluate equation 4.121. This approach works best for a system composed of a dilute, randomly dispersed arrangement of discrete scattering particles that have a definite shape. Also, if the distribution of interparticle distance can be stated, interparticle interference effects can be included in this approach. The model approach is used in deriving equations 4.51 and 4.55. More specifically, this approach is the basis of the electron and the x-ray scattering techniques for determining the structure of small molecules. Van de Hulst [30] and Chu [31] have covered in detail the effects of... 

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