Properties of probability-density functions

The basic definition of a probability-density function is given in Section 10.1.1. From P(vX the average of the random variable is readily calculated as 

This is also the first moment of P(v) about zero because the nth moment about zero is defined as 

The mean-square value, is the second moment about zero. The variance is the second moment about the mean, also termed the second central moment, and conventionally is denoted by 

A result of the definitions, the meaning of which should be clear in spite of the somewhat imprecise notation, is Bayes s theorem, P(i i i 2) ( 2) = P(v2 v )P(v in which all four Fs are different functions. 

Conditioning often is needed in turbulent combustion. For example, often there is interest in flows (for example, jets and wakes) that are partly turbulent and partly nonturbulent (in that the flow is partly irrotational), 

Conditioning often is needed in turbulent combustion. For example, often there is interest in flows (for example, jets and wakes) that are partly turbulent and partly nonturbulent (in that the flow is partly irrotational), and the conditioning may be that rotational fluid be present. These flows are said to exhibit intermittency of the turbulence an intermittency function may be defined as a stochastic function having a value /(x, t) = 0 in irrotational flow and /(x, t) = 1 in rotational flow, and conditioning on / = 1 may be desirable for a number of purposes. We may decompose a probability-density function as 

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State vector, specification of, 493 Stationarity property of probability density functions, 136 Stationary methods, 60 Statistical independence, 148 Statistical matrix, 419 including description of "mixtures, 423... 

By definition, P(v) possesses the necessary nonnegative and normalization properties of probability-density functions. It is especially useful in connection with the Favre-averaged formulation of the conservation equations, since corresponding averages are obtained from P(v) by the usual rules for averaging. Thus, for the v and v" defined above equation (2),... 

Integration of eq. (6) over all times and the property of probability density functions provides a check on the tracer experiment ... 

One of the most important properties of Fourier transforms and, consequently, of characteristic functions, is their invertibility. Given a characteristic function M, one can calculate the probability density function p by means of the inversion formula... 

The properties of joint distribution functions can be stated most easily in terms of their associated probability density functions. The n + mth order joint probability density function px. . , ( > ) is defined by the equation... 

Some of the most important properties of higher-order probability density functions are the following 28... 

Equation (3-104) (sometimes called the stationarity property of a probability density function) follows from the definition of the joint distribution function upon making the change of variable t = t + r... 

In other words, if we assume that the counting function N(t) has statistically independent increments (Eq. (3-237)), and has the property that the probability of a single jump occurring in a small interval of length h is approximately nh but the probability of more than one jump is zero to within terms of order h, (Eq. (3-238)), then it can be shown 51 that its probability density functions must be given by Eq. (3-231). It is the existence of theorems of this type that accounts for the great... 

A third measure of location is the mode, which is defined as that value of the measured variable for which there are the most observations. Mode is the most probable value of a discrete random variable, while for a continual random variable it is the random variable value where the probability density function reaches its maximum. Practically speaking, it is the value of the measured response, i.e. the property that is the most frequent in the sample. The mean is the most widely used, particularly in statistical analysis. The median is occasionally more appropriate than the mean as a measure of location. The mode is rarely used. For symmetrical distributions, such as the Normal distribution, the mentioned values are identical. 

The chemical properties of particles are assumed to correspond to thermodynamic relationships for pure and multicomponent materials. Surface properties may be influenced by microscopic distortions or by molecular layers. Chemical composition as a function of size is a crucial concept, as noted above. Formally the chemical composition can be written in terms of a generalized distribution function. For this case, dN is now the number of particles per unit volume of gas containing molar quantities of each chemical species in the range between ft and ft + / ,-, with i = 1, 2,..., k, where k is the total number of chemical species. Assume that the chemical composition is distributed continuously in each size range. The full size-composition probability density function is... 

Table F.l Some Laplace transform properties and pairs of functions used as probability density functions for semi-Markov modeling.

There are some general properties of all distribution functions irrespective of the type of distribution. For example F x) 0 is a practical constraint because negative probabilities have no meaning. Typically the probability distribution functions are normalized, i.e., the density function integrated over the entire domain of the density function has a total area under the curve of one ... 

Because this analysis is required at each point X, we drop the subscript 0, and discuss the properties of SP(X) in general. The function Sf (X), called the probability density function for X, is plotted as a continuous function above the X-axis. It is defined for all points in an interval (a, b) whose end points depend on the nature of X. In some cases the end points can include -l-oo,-oo, or both. Note that Sf (X) has physical dimensions of X since the product Sf (X)dX must be dimensionless. 

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