Moment of a distribution

Moments of a distribution often provide information that can be used to characterize particulate matter. Theyth moment of the population density function n is defined as... 

Generating functions are used in calculating moments of distributions for power series expansions. In general, the nth moment of a distribution,/fxj is E x ") = lx" f x) dx, where the integration is over the domain of x. (If the distribution is discrete, integration is replaced by summation.)... 

The second order central moment is used so frequently that it is very often designated by a special symbol o2 or square root of the variance, o, is usually called the standard deviation of the distribution and taken as a measure of the extent to which the distribution is spread about the mean. Calculation of the variance can often be facilitated by use of the following formula, which relates the mean, the variance, and the second moment of a distribution... 

It can be shown, however, that, under certain conditions, knowledge of all the moments of a distribution determines the distribution uniquely. 

Equation (15.39) allows moments of a distribution to be calculated from the Laplace transform of the dilferential distribution function without need for finding f t). It works for any f t). The necessary algebra for the present case is formidable, but finally gives the desired relationship ... 

The electrostatic moments of a distribution p(r) in terms of the clmp are given by... 

Factorial Moments. For finding the moments of a distribution such as the Poisson, a useful device is the factorial moment. (The Poisson distribution is given in Example 3.1.) The density is... 

It has been shown that, if all the moments of a distribution are known, then the distribution may under certain conditions be expressed as a series in terms of the variable and the moments. The expansion is given by... 

As an aside, it should be noted that E(X) and E(X2) are referred to as the first and second moments of a distribution and that E(Xk) is referred to as the kth moment of a pdf. Returning to the mean and variance, the mean of an exponential distribution (which is useful for modeling the time to some event occurring) having pdf... 

In this part we will explain a method that we extensively used to describe stochastic dynamical systems. It is based on the dynamics of the moments of a distribution. We applied it successfully to a variety of globally coupled systems. Advantages of the method are simple applicability and quick numerical investigations. Let us consider a globally coupled stochastic system that is described by the following set of Langevin equations ... 

The Cumulant expansion is a series method for approximating the moments of a distribution. It is often used to fit a non-Gaussian or anharmonic interatomic distance distribution in EXAFS spectra. 

Some of the most important properties that a quantum mechanical calculation provides are the electric multipole moments of the molecule. The electric multipoles reflect the distribution of charge in a molecule. The simplest electric moment (apart from the total net charge on the molecule) is the dipole. The dipole moment of a distribution of charges located at positions r, is given by ij r/. If there are just two charges +q and -q separated by a distance... 

First moment of a distribution. Ml of a photon distribution versus time represents the average arrival time of the photons. 

One way to parameterize the absorbance spectrum e(v) is by calculation of the moments of the millimolar extinction coefficient distribution. The nth moment of a distribution is given by... 

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... 

Mean and variance can be derived from the moments of a distribution. In general, the rth central moment calculated about the mean is... 

This distribution of molecular masses and sizes can now be routinely quantified for all soluble polymers by gel permeation chromatography (1). While this distribution is useful both in practice and in theory, many properties of the polymer sample depend on a single middle value of the distribution. There are, however, several ways to reduce the distribution to a middle value. Each of these reductions is important because they correlate with or predict a certain subset of physical or chemical properties of the polymer. The common averages of a polymer molecular mass distribution are number (m=1), viscosity (m=1 +a ), weight (m=2), and z or zeta average (m=3). These "averages" are actually ratios of the m- moment of the molecular mass distribution to the preceding moment in the above list. The moments of a distribution are fundamental properties of any distributed variable and are covered in detail in reference 2. 

Special applications of the above formula are the moments of a distribution. The nth moment is defined as follows ... 

Mean, avers e, expected value The first moment of a distribution the integral of x with respect to its density function f x). 

Variance The second moment of a distribution about its mean measures the dispersion of the distribution. 

Angle brackets () are used to indicate the moments, also called the expectation values or averages, of a distribution function. For a probability distribution the zeroth moment always equals one, because the sum of the probabilities equals one. The first moment of a distribution function (n = 1 in Equation (1.32)) is called the mean, average, or expected value. For discrete functions... 

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