Random process mean value

A random number (between 0 and 1) is picked, and the associated value of gross reservoir thickness (T) is read from within the range described by the above distribution. The value of T close to the mean will be randomly sampled more frequently than those values away from the mean. The same process is repeated (using a different random number) for the net-to-gross ratio (N/G). The two values are multiplied to obtain one value of net sand thickness. This is repeated some 1,000-10,000 times, with each outcome being equally likely. The outcomes are used to generate a distribution of values of net sand thickness. This can be performed simultaneously for more than two variables. 

The mean values of the. (t) are zero and each is assumed to be stationary Gaussian white noise. The linearity of these equations guarantees that the random process described by the a. is also a stationary Gaussian-... 

For the usual accurate analytical method, the mean f is assumed identical with the true value, and observed errors are attributed to an indefinitely large number of small causes operating at random. The standard deviation, s, depends upon these small causes and may assume any value mean and standard deviation are wholly independent, so that an infinite number of distribution curves is conceivable. As we have seen, x-ray emission spectrography considered as a random process differs sharply from such a usual case. Under ideal conditions, the individual counts must lie upon the unique Gaussian curve for which the standard deviation is the square root of the mean. This unique Gaussian is a fluctuation curve, not an error curve in the strictest sense there is no true value of N such as that presumably corresponding to a of Section 10.1—there is only a most probable value N. 

The important point we wish to re-emphasize here is that a random process is specified or defined by giving the values of certain averages such as a distribution function. This is completely different from the way in which a time function is specified i.e., by giving the value the time function assumes at various instants or by giving a differential equation and boundary conditions the time function must satisfy, etc. The theory of random processes enables us to calculate certain averages in terms of other averages (known from measurements or by some indirect means), just as, for example, network theory enables us to calculate the output of a network as a function of time from a knowledge of its input as a function of time. In either case some information external to the theory must be known or at least assumed to exist before the theory can be put to use. 

Owing to the complexity of turbulent flow, it is usually treated as if it were a random process. In addition, it is usually adequate to calculate mean values of flow quantities, but as will be seen these are not always as simple as might be expected. The instantaneous value of the velocity... 

At high Reynolds number, the velocity U(x, t) is a random field, i.e., for fixed time t = t the function U(x, D varies randomly with respect to x. This behavior is illustrated in Fig. 2.1 for a homogeneous turbulent flow. Likewise, for fixed x = x lJ(x. t) is a random process with respect to t. This behavior is illustrated in Fig. 2.2. The meaning of random in the context of turbulent flows is simply that a variable may have a different value each time an experiment is repeated under the same set of flow conditions (Pope 2000). It does not imply, for example, that the velocity field evolves erratically in time and space in an unpredictable fashion. Indeed, due to the fact that it must satisfy the Navier-Stokes equation, (1.27), U(x, t) is differentiable in both time and space and thus is relatively smooth. ... 

Why is any of this of interest If it is known that some data are normally distributed and one can estimate p and a, then it is possible to state, for example, the probability of finding any particular result (value and uncertainty range) the probability that future measurements on the same system would give results above a certain value and whether the precision of the measurement is lit for purpose. Data are normally distributed if the only effects that cause variation in the result are random. Random processes are so ubiquitous that they can never be eliminated. However, an analyst might aspire to reducing the standard deviation to a minimum, and by knowing the mean and standard deviation predict their effects on the results. 

To ensure that each pixel is correctly exposed, a minimum number of electrons must strike each pixel. Since electron emission is a random process, the actual number of electrons striking each pixel, n, will vary in a random manner about a mean value, n. Adapting the signal-to-noise analysis found in Schwartz (1959) to the case of binary exposure of a resist, one can show straightforwardly that the probability of error for large values of the mean number of electrons/pixel it is /[( /2) ] 2. This leads to the following table of probability of error of exposure ... 

The statistics of processes such as radioactive decay and emission of light that produce a flux of particles or distributive polymerase enzymes that add residues at random to growing polymer chains obey the Poisson distribution (see Chapter 14). The number of particles measured per unit time or the number of residues added to a particular chain varies about the mean value x according to equation 6.41. 

Radioactive decay with emission of particles is a random process. It is impossible to predict with certainty when a radioactive event will occur. Therefore, a series of measurements made on a radioactive sample will result in a series of different count rates, but they will be centered around an average or mean value of counts per minute. Table 1.1 contains such a series of count rates obtained with a scintillation counter on a single radioactive sample. A similar table could be prepared for other biochemical measurements, including the rate of an enzyme-catalyzed reaction or the protein concentration of a solution as determined by the Bradford method. The arithmetic average or mean of the numbers is calculated by totaling all the experimental values observed for a sample (the counting rates, the velocity of the reaction, or protein concentration) and dividing the total by the number of times the measurement was made. The mean is defined by Equation 1.1. 

In addition it is now time to think about the two assumption models, or types of analysis of variance. ANOVA type 1 assumes that all levels of the factors are included in the analysis and are fixed (fixed effect model). Then the analysis is essentially interested in comparing mean values, i.e. to test the significance of an effect. ANOVA type 2 assumes that the included levels of the factors are selected at random from the distribution of levels (random effect model). Here the final aim is to estimate the variance components, i.e. the variance fractions with respect to total variance caused by the samples taken or the measurements made. In that case one is well advised to ensure balanced designs, i.e. equally occupied cells in the above scheme, because only then is the estimation process straightforward. 

Analysis of variance in general serves as a statistical test of the influence of random or systematic factors on measured data (test for random or fixed effects). One wants to test if the feature mean values of two or more classes are different. Classes of objects or clusters of data may be given a priori (supervised learning) or found in the course of a learning process (unsupervised learning see Section 5.3, cluster analysis). In the first case variance analysis is used for class pattern confirmation. 

The major objective in SPC is to use process data and statistical techniques to determine whether the process operation is normal or abnormal. The SPC methodology is based on the fundamental assumption that normal process operation can be characterized by random variations around a mean value. The random variability is caused by the cumulative effects of a number of largely unavoidable phenomena such as electrical measurement noise, turbulence, and random fluctuations in feedstock or catalyst preparation. If this situation exists, the process is said to be in a state of statistical control (or in control), and the control chart measurements tend to be normally distributed about the mean value. By contrast, frequent control chart violations would indicate abnormal process behavior or an out-of-control situation. Then a search would be initiated to attempt to identify the assignable cause or the. special cause of the abnormal behavior... 

White noise is by definition a random signal with a flat power spectral density (i.e., the noise intensity is the same for all frequencies or all times, of course, within a finite range of frequencies or times). A time-random process u>w(t) is white in the time range a < t < b if and only if its mean value is zero ... 

The exit variables that present an indirect relation with the particularities of the process evolution, denoted here by Cj,l = 1,Q, are recognized as intermediary variables or as exit control variables. The exit process variables that depend strongly on the values of the independent process variables are recognized as dependent process variables or as process responses. These are denoted by y ,i = 1, P. When we have random inputs in a process, each y exit presents a distribution around a characteristic mean value, which is primordially determined by the state of all independent process variables Xj, j = 1, N. Figure 1.1 (b), shows an abstract scheme of a tangential filtration unit as well as an actual or concrete picture. 

The decisions made by the computer concerning the pressure of the pump-flow rate dependence and of the flow rate of the fresh suspension, are controlled by the micro-device of the execution system (ES). It is important to observe that the majority of the input process variables are not easily and directly observable. As a consequence, a good technological knowledge is needed for this purpose. If we look attentively at the xj — X5 input process variables, we can see that their values present a random deviation from the mean values. Other variables such as pump exit pressure and flow rate (xg,X7) can be changed with time in accordance with technological considerations. 

The models associated to a process with no randomly distributed input variables or parameters are called rigid models. If we consider only the mean values of the parameters and variables of one model with randomly distributed parameters or input variables, then we transform a non-deterministic model into a rigid model. 

We have to notice that, for different X(t) values, we associate different values for the elements of the matrix of transition probabilities. When the movement randomly changes the value of X into a value around A, Eq. (4.91) is formulated with expressions giving the probability of process X(t) at different states. The infinitesimal operator [Qf] ([Qf] = Q by function f) is defined as the temporary derivative of the mean value of the stochastic process for the case when the process evolves randomly... 

In the case where X(t) or X(t, e) corresponds to a diffusion process (the stochastic process is continuous), it can be demonstrated that Q is a second order elliptic operator [4.39- 4.42]. The solution of the equation, which defines the random evolution, is given by a formula that yields 0(r,t). In this case, if we can consider that 6jjj(t, X) is the mean value of X(t)(which depends on the initial value of Xq), then, we can write the following equation ... 

A real process is frequently influenced by non-commanded and non-controlled small variations of the factors and also by the action of other random variables (Fig. 5.1). Consequently, when the experiments are planned so as to identify coefficients Po pj etc, they will apparently show different collected data. So, each experiment will have its own Pq P etc. coefficients. In other words, each coefficient is a characteristic random variable, which is observable by its mean value and dispersion. 

The number of cycles for each stage must be thoroughly selected because then the interest is to observe the small changes occurring simultaneously with the permanent random fluctuations in the process output. The data from a cycle are transferred to the next cycle to complete the new phase by calculation of the mean values and variances. It is well known that the errors in the mean value of n independent observations are s/n smaller than the error of an isolated measure. Therefore, this fact sustains the transfer of data from one cycle to the next one. 

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