Probability density function value

FIGURE 5.5 Bayesian posterior normal probability density function values for SSD for cadmium and its Bayesian confidence limits 5th, 50th, and 95th percentiles (black) and Bayesian posterior probability density of the HC5 (gray). 

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. 

Fig. 10.19 The probability density of the extreme value distribution typical of the MSP scores for random sequena The probability that a random variable with this distribution has a score of at least x is given by 1 - exp[-e -where u is the characteristic value and A is the decay constant. The figure shows the probability density function (which corresponds to the function s first derivative) for u = 0 and A = 1.

The probability density function of u is shown for four points in Fig. 11.16, two points in the wall jet and two points in the boundary layer close to the floor. For the points in the wall jet (Fig. 11.16<2) the probability (unction shows a preferred value of u showing that the flow has a well-defined mean velocity and that the velocity is fluctuating around this mean value. Close to the floor near the separation at x/H = I (Fig. 11.16f ) it is hard to find any preferred value of u, which shows that the flow is irregular and unstable with no well-defined mean velocity and large turbulent intensity. From Figs. 11.15 and 11.16 we can see that LES gives us information about the nature of the turbulent fluctuations that can be important for thermal comfort. This type of information is not available from traditional CFD using models. 

The conditional probability density functions defined by Eq. (3-170) are joint probability density functions for fixed values of xn... 

We shall conclude this section by investigating the very interesting behavior of the probability density functions of Y(t) for large values of the parameter n. First of all, we note that both the mean and the covariance of Y(t) increase linearly with n. Roughly speaking, this means that the center of any particular finite-order probability density function of Y(t) moves further and further away from the origin as n increases and that the area under the density function is less and less concentrated at the center. For this reason, it is more convenient to study the normalized function Y ... 

X 10 years old, this implies that the content of the reservoir today is about half of what it was when the Earth was formed. The probability density function of residence time of the uranium atoms originally present is an exponential decay function. The average residence time is 6.5 x 10 years. (The average value of... 

Luo and Domfeld [110] introduced a fitting parameter H , a d5mamical" hardness value of the wafer surface to show the chemical effect and mechanical effect on the interface in their model. It reflects the influences of chemicals on the mechanical material removal. It is found that the nonlinear down pressure dependence of material removal rate is related to a probability density function of the abrasive size and the elastic deformation of the pad. 

This conditional cdf is a function not only of the data configuration (N locations ly. i l,, N) but also of the N data values (pi, i l,, N) Its derivative with regard to the argument z is the conditional probability density function (pdf) and is denoted by ... 

Table 2.3 is used to classify the differing systems of equations, encountered in chemical reactor applications and the normal method of parameter identification. As shown, the optimal values of the system parameters can be estimated using a suitable error criterion, such as the methods of least squares, maximum likelihood or probability density function. 

The application of optimisation techniques for parameter estimation requires a useful statistical criterion (e.g., least-squares). A very important criterion in non-linear parameter estimation is the likelihood or probability density function. This can be combined with an error model which allows the errors to be a function of the measured value. A simple but flexible and useful error model is used in SIMUSOLV (Steiner et al., 1986 Burt, 1989). 

If the mathematical model of the process under consideration is adequate, it is very reasonable to assume that the measured responses from the i,h experiment are normally distributed. In particular the joint probability density function conditional on the value of the parameters (k and ,) is of the form,... 

The distribution of stretches can be quantified in terms of the probability density function F (X)=dN(X)/dX, where dN(A) is the number of points that have values of stretching between A and (X+dX) at the end of period n. Another possibility is to focus on the distribution of log A. In this case we define the measure //n(logA)=dA(logA)/d(logA). 

Human parameters Value Probability density function Units Reference... 

The total suspended particles value considered as input was 1.24 x 10 4 g m 3 whereas the global solar radiation was set at 158.95. Table 14 shows the input parameters used in the form of probability density function(PDF) and which allow the probabilistic analysis and sensitivity analysis in terms of simulation outcomes. 

The statistical fundamentals of the definition of CV and LD are illustrated by Fig. 7.8 showing a quasi-three-dimensional representation of the relationship between measured values and analytical values which is characterized by a calibration straight line y = a + bx and their two-sided confidence limits and, in addition (in z-direction) the probability density function the measured values. 

Fig. 7.8. Schematic three-dimensional representation of a calibration straight line of the form y = a + bx with the limits of its two-sided confidence interval and three probability density function (pdf) p(y) of measured values y belonging to the analytical values (contents, concentrations) X(A) = 0 (A), x = x(B) (B) and X(q = ld (C) yc is the critical value of the measurement quantity a the intercept of the calibration function yBL the blank x(B) the analytical value belonging to the critical value yc (which corresponds approximately to Kaiser s a3cr-limit ) xLD limit of detection... 

In a turbulent flow, the local value (i.e., at a point in space) of the mixture fraction will behave as a random variable. If we denote the probability density function (PDF) of by f - Q where 0 < ( < 1, the integer moments of the mixture fraction can be found by integration ... 

Figure 4.1 Relationship between the probability density function f x) of the continuous random variable X and the cumulative distribution function F(x). The shaded area under the curve f(x) up to x0 is equal to the value of f x) at x0. 

Figure 4.6 Calculated probability density function of 5lsO values in rainwater (see text for assumptions and parameters).

FIGURE 2.4 Probability density function of the uniform distribution (left), and the logit-transformed values as solid line and the standard normal distribution as dashed line (right). 

Figure 6. Probability density functions of the skews Sta ( ) and Sac ( ) values computed from the intronic sequences of the 14,854 intron-containing genes after removing repeated sequences, (a) Sense genes (b) antisense genes.

Figure 13. Probability density functions of the skews Sja (a), Sac (b)- S = Sja + Sec (c) values computed in nonoverlapping 1-kbp windows from the DNA sequences of the 22 human autosomal chromosomes. Symbols have the following meaning (o) native sequences and ( ) repeat-masqued sequences.

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... 

Scheme 4.2 Bond energy as a function of hydrogen position (black solid line), assuming identical pff, values for the donor and acceptor, relative to the lowest vibrational energy level of the hydrogen atom (highlighted by a dotted line), (a) A standard, symmetric hydrogen bond (b) the corresponding low-barrier hydrogen bond (LBHB). The red line represents the probability density function [27, 28].

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...

Raising to an inverse power has the effect of converting minima into maxima, and the division by the atomic valence gives p, a value of 1.0 at an ideal location. If N is set equal to 16, the maxima become quite sharp and the resultant p, map contains peaks that, under suitable conditions, resemble the probability density function for the atom at room temperature as shown for... 

A very important probability distribution is the normal or Gaussian distribution (after the German mathematician, Karl Friedrich Gauss, 1777-1855). The normal distribution has the same value for the mean, median and mode. The equation describing this distribution (the probability density function)... 

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