Probability density function, range

When providing input for the STOMP calculation a range of values of porosity (and all of the other input parameters) should be provided, based on the measured data and estimates of how the parameters may vary away from the control points. The uncertainty associated with each parameter may be expressed in terms of a probability density function, and these may be combined to create a probability density function for STOMP. 

It is common practice within oil companies to use expectation curves to express ranges of uncertainty. The relationship between probability density functions and expectation curves is a simple one. 

Nagode, M. and Fajdiga, M. 1998 A General Multi-Modal Probability Density Function Suitable for the Rainflow Ranges of Stationary Random Processes. Int. Journal of Fatigue, 20(3), 211-223. 

A model must be introduced to simulate fast chemical reactions, for example, flamelet, or turbulent mixer model (TMM), presumed mapping. Rodney Eox describes many proposed models in his book [23]. Many of these use a probability density function to describe the concentration variations. One model that gives reasonably good results for a wide range of non-premixed reactions is the TMM model by Baldyga and Bourne [24]. In this model, the variance of the concentration fluctuations is separated into three scales corresponding to large, intermediate, and small turbulent eddies. 

In a first simple model for target detection it is assumed that the background clutter can be described by a statistical model in which the different range cells inside the sliding window contain statistically independent identically exponentially distributed (iid) random variables X i,..., X/v. The probability density function (pdf) of exponentially distributed clutter variables is fully described by the equation ... 

Both the number and weight basis probability density functions of final product crystals were found to be expressed by a %2-function, under the assumption that the CSD obtained by continuous crystallizer is controlled predominantly by RTD of crystals in crystallizer, and that the CSD thus expressed exhibits the linear relationships on Rosin-Rammler chart in the range of about 10-90 % of the cumulative residue distribution. 

Formally, suppose we have a random variable, jc, which has measurements over the range a to b. Also, assume that the probability density function of x can be written as p(x). In addition, assume a second function g, such that g(x ) p(x) =J x). For example, gix) could represent a dose-response function on concentration and p(x) is the probability density function of concentration. The expected value (which is the most likely value or the mean value) of g(J ) isp... 

Probability density function (PDF) The PDF is referred to as the probability function or the frequency function. For continuous random variables, that is, the random variables that can assume any value within some defined range (either finite or infinite), the probability density function expresses the probability that the random variable falls within some very small interval. For... 

Fig. 23. Comparison of classification by probability density functions and by Mahalanobis distance. Univariate case. Values in the range between the dotted lines are classified into class b...

The normal, or Gaussian, distribution occupies a central place in statistics and measurement. Its familiar bell-shaped curve (the probability density function or pdf, figure 2.1) allows one to calculate the probability of finding a result in a particular range. The x-axis is the value of the variable under consideration, and the y-axis is the value of the pdf. 

Equation (10.40) is sometimes found in a simpler form at the cost of hiding the complexity of the terms involved. This form is based on the introduction of a probability density function for the reaction coordinate and the associated potential of mean force, in contrast to previously, where we considered the probability density of a particular arrangement of n atoms. Let I l(Q )d,Q be the probability of finding the reaction coordinate in the range Q, Q + dQ. Then, from equilibrium statistical mechanics (see Appendix A.2), the probability density function I l(Q ) is given by... 

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... 

In an unmodified Monte Carlo method, simple random sampling is used to select each member of the 777-tuple set. Each of the input parameters for a model is represented by a probability density function that defines both the range of values that the input parameters can have and the probability that the parameters are within any subinterval of that range. In order to carry out a Monte Carlo sampling analysis, each input is represented by a cumulative distribution function (CDF) in which there is a one-to-one correspondence between a probability and values. A random number generator is used to select probability in the range of 0-1. This probability is then used to select a corresponding parameter value. 

Probability density function (pdf) Indicates the relative likelihood of the different possible values of a random variable. For a discrete random variable, say X, the pdf is a function, say /, such that for any value x, /(x) is the probability that X = X. For example, if X is the number of pesticide applications in a year, then /(2) is the probability density function at 2 and equals the probability that there are two pesticide applications in a year. For a continuous random variable, say Y, the pdf is a function, say g, such that for any value y, g(y) is the relative likelihood that Y = y,0 < g y), and the integral of g over the range of y from minus infinity to plus infinity equals 1. For example, if Y is body weight, then g(70) is the probability density function for a body weight of 70 and the relative likelihood that the body weight is 70. Furthermore, if g 70)/g(60) = 2, then the body weight is twice as likely to be 70 as it is to be 60 (Sielken, Ch. 8). 

With turbulent combustion viewed as a random (or stochastic) process, mathematical bases are available for addressing the subject. A number of textbooks provide introductions to stochastic processes (for example, [55]). In turbulence, any stochastic variable, such as a component of velocity, temperature, or the concentration of a chemical species, which we might call v, is a function of the continuous variables of space x and time t and is, therefore, a stochastic function. A complete statistical description of a stochastic function would be provided by a probability-density functional, tf, defined by stating that the probability of finding the function in a small range i (x, t) about a particular function v(x, t) is [t (x, t)]<3t (x, t) ... 

An avenue that has received exploration is the development of equations for evolution of probability-density functions. If, for example, attention is restricted entirely to particular, fixed values of x and t, then the variable whose value may be represented by v becomes a random variable instead of a random function, and its statistics are described by a probability-density function. The probability-density function for v may be denoted by P(v where P(v) dv is the probability that the random variable lies in the range dv about the value v. By definition P(v) > 0, and P(v) dv = 1, One approach to obtaining an equation of evolution for P(v) is to introduce the ensemble average of a fine-grained density, as described by O Brien in [27], for example another is formally to perform suitable integrations in... 

The Monte Carlo method is a well-established approach used in characterizing uncertainty that can be used to incorporate ranges of data (distributions) into calculations. The Monte Carlo method involves choosing values from a random selection scheme drawn from probability density functions based on a range of data that characterize the parameters of interest. Monte Carlo analysis can be selectively used to generate input parameters and mixed with point estimates, as appropriate, to calculate risk. 

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