Probability distribution results

To apply Equation (1.14), recorded data are arranged into arrays either from the highest to the lowest or from the lowest to the highest. The number of values above a given element and including the element is counted and the probability equation applied to each individual element of the array. Because the number of values above and at a particular element is a sum, this application of the equation is, in effect, an application of the probability of the union of events. The probability is called cumulative, or union probability. After all union probabilities are calculated, an array of probability distribution results. This method is therefore called probability distribution analysis. This method will be illustrated in the next example. 

From the probability distributions for each of the variables on the right hand side, the values of K, p, o can be calculated. Assuming that the variables are independent, they can now be combined using the above rules to calculate K, p, o for ultimate recovery. Assuming the distribution for UR is Log-Normal, the value of UR for any confidence level can be calculated. This whole process can be performed on paper, or quickly written on a spreadsheet. The results are often within 10% of those generated by Monte Carlo simulation. 

We will refer to this model as to the semiclassical QCMD bundle. Eqs. (7) and (8) would suggest certain initial conditions for /,. However, those would not include any momentum uncertainty, resulting in a wrong disintegration of the probability distribution in g as compared to the full QD. Eor including an initial momentum uncertainty, a Gaussian distribution in position space is used... 

Bell-shaped probability distribution curve for measurements and results showing the effect of random error. 

The distribution of the results of an analysis around a central value is often described by a probability distribution, two examples... 

Age appears to improve the flowability of certain materials. This is probably the result of particle-surface oxidation, more even moisture distribution, and the rounding of particle corners caused by handhng. 

Figure 2.5-1 illustrates the fact that probabilities are not precisely known but may be represented by a "bell-like" distribution the amplitude of which expresses the degree of belief. The probability that a system will fail is calculated by combining component probabilities as unions (addition) and intersection (multiplication) according to the system logic. Instead of point values for these probabilities, distributions are used which results in a distributed probabilitv of system fadure. This section discusses several methods for combining distributions, namely 1) con olution, 2i moments method, 3) Taylor s series, 4) Monte Carlo, and 5) discrete probability distributions (DPD). 

In the final stage, when the dimethochloride of either Aim thyldesbisneo-strychnidine or that of dimethyldesstrychnidine-D is heated with sodium methoxide in alcohol N (6) is eliminated as trimethylamine and there is formed a mixture of the two desazostrychnidines, a and b, of which the first is amorphous but yields a crystalline methiodide, m.p. 154-5°, and the second is crystalline, m.p. 109-110°, giving a methiodide, m.p. 105-6°. Each yields a hexahydro-derivative, which may be a mixture of stereo-isomerides, and the differenee between the forms a- and h- is probably the result of dissimilar distribution of the three ethylenie linkages thus indi-... 

Where f(x) is tlie probability of x successes in n performances. One can show that the expected value of the random variable X is np and its variance is npq. As a simple example of tlie binomial distribution, consider tlie probability distribution of tlie number of defectives in a sample of 5 items drawn with replacement from a lot of 1000 items, 50 of which are defective. Associate success with drawing a defective item from tlie lot. Tlien the result of each drawing can be classified success (defective item) or failure (non-defective item). The sample of items is drawn witli replacement (i.e., each item in tlie sample is relumed before tlie next is drawn from tlie lot tlierefore the probability of success remains constant at 0.05. Substituting in Eq. (20.5.2) tlie values n = 5, p = 0.05, and q = 0.95 yields... 

In its more advanced aspects, kinetic theory is based upon a description of the gas in terms of the probability of a particle having certain values of coordinates and velocity, at a given time. Particle interactions are developed by the ordinary laws of mechanics, and the results of these are averaged over the probability distribution. The probability distribution function that is used for a given macroscopic physical situation is determined by means of an equation, the Boltzmann transport equation, which describes the space, velocity, and time changes of the distribution function in terms of collisions between particles. This equation is usually solved to give the distribution function in terms of certain macroscopic functions thus, the macroscopic conditions imposed upon the gas are taken into account in the probability function description of the microscopic situation. 

Reactions of Complex Ions. For reactions of systems containing H2 or HD the failure to observe an E 1/2 dependence of reaction cross-section was probably the result of the failure to include all products of ion-molecule reaction in the calculation of the experimental cross-sections. For reactions of complex molecule ions where electron impact ionization probably produces a distribution of vibrationally excited states, kinetic energy transfer can readily open channels which yield products obscured by primary ionization processes. In such cases an E n dependence of cross-section may be determined frequently n = 1 has been found. 

Analogous experiments using electrons instead of photons have been carried out with the same results. Electrons passing through a system with double slits produce an interference pattern. If a detector determines through which slit each electron passes, then the interference pattern is not observed. As with the photon, the electron exhibits both wave-like and particle-like behavior and its location on a detection screen is randomly determined by a probability distribution. 

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