Numerical values of the probability integrals

What is the best representative value of a series of measurements affected with errors of observations  

How closely does the arithmetical mean approximate to the absolute truth To illustrate, we may use the results of Crookes model research on the atomic weight of thallium (Phil. Trans., 163, 277, 1874). Crooke s determination of this constant gave 203-628 203 632 203 636 203 638 203 6391 

203 642 203 644 203 649 203 650 203 666 JMean 203 642 The arithmetical mean is only one of an infinite number of possible values of the atomic weight of thallium between the extreme limits 203 628 and 203 666. It is very probable that 203 642 is not the true value, but it is also very probable that 203 642 is very near to the true value sought. The question How near cannot be answered. Alter the question to What is the probability that the truth is comprised between the limits 203 642 + x t and the answer may be readily obtained however small we choose to make the number x. 

suppose that the absolute measure of precision, h, of the arithmetical mean is known. Table X. gives the numerical values of the probability integral 

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Extensive tables of numerical values of the integral P in Equation [8.15] are given in any statistics textbook it is important to understand that P , represents the probability that, for a very large number of measurements that conform to the mathematical model described by the normal distribution G (z), a particular value for z (and thence for x = p -I- z(j) will fall within a range m on either side of z = 0 (i.e., of X = p). Figure 8.6 shows values of P calculated for small integral values of m and also one example of the converse case, i.e., a value of m shown to correspond to a user-selected value of P (= 0.5 in this case, i.e., to the case where 50 % of all measurements will fall within the calculated range of z and thus x). 

The integral of the Gaussian distribution function does not exist in closed form over an arbitrary interval, but it is a simple matter to calculate the value of p(z) for any value of z, hence numerical integration is appropriate. Like the test function, f x) = 100 — x, the accepted value (Young, 1962) of the definite integral (1-23) is approached rapidly by Simpson s rule. We have obtained four-place accuracy or better at millisecond run time. For many applications in applied probability and statistics, four significant figures are more than can be supported by the data. 

Fig. 35). The potential energy curves and the transition dipole moment are taken from [117]. The time evolution of the populations on the ground and excited states is shown in Fig. 36 More than 86% of the initial state is excited to the B state within the period shorter than a few femtoseconds. The integrated total transition probability V given by Eq. (173) is P = 0.879, which is in good agreement with the value 0.864 obtained by numerical solution of the original coupled Schroedinger equations. This means that the population deviation from 100% is not due to the approximation, but comes from the intrinsic reason, that is, from the spread of the wavepacket. Note that the LiH molecule is one of the...

Figure 3 gives 1/F as a function of r for some probable values of the parameters involved and with W obtained 1 numerical integration of Eq. (IS). The more important features of Eq. (22) are evident from the figure. 

The extraction of numerical values for the local densities of state at the Fermi energy from NMR resonance position and relaxation rate requires of course a number of hypotheses. Some of them (such as knowledge of the resonance position corresponding to zero total shift the breakup of the density of states into parts of different symmetry, etc.) already come into play when we try to parameterize data for the bulk metal [58]. Here we mention only the additional ones used to go to the local version of the equations. It is assumed that the hyperfine fields and exchange integrals are a kind of atomic properties that do not vary when the atom is put in one environment or another, whether it is deep inside the particle or on its surface. The approximation is probably reasonable when the atomic volume stays approximately... 

The simulated spectrum is computed by the use of eq. (25), wherein the integrals are converted into discrete sums. It is clear from (25) that, in particular, one needs to know the resonant field values for the various transitions, as well as their transition probabilities for numerous orientations of the external magnetic field over the unit sphere over the unit sphere. A considerable saving of computer time can be accomplished if one uses numerical techniques to minimize the number of required diagonalizations of the SH matrix in the brute-force method. That is, when one uses the known resonant-field value at angle (0,(p) to calculate the one at an infinitesimally close orientation, (0 -i- 80, (p + 8(p), known as the method of homo-... 

For negative x values, the value 1.0-yp(-x)ean be evaluated due to the symmetry of the probability function. There is some ehoiee in the seleetion of values of m and q to use in this equation. One would expeet fliat flie fitting aeeuraey increases as m increases and perhaps also as q inereases, within some limits. From various experiments, it can be found that good results ean be aehieved with eom-binations such as m = 4, q = 4 and m = 6, q=16. In order to experiment with approximating the probability function with equations of this form one must first have a means of accurately evaluating the integral for some finite number of points. For this work, this will be done by use of the numerical integration tech-... 

Later, Kuppermann and Belford (1962a, b) initiated computer-based numerical solution of (7.1), giving the space-time variation of the species concentrations from these, the survival probability at a given time may be obtained by numerical integration over space. Since then, this method has been vigorously followed by others. John (1952) has discussed the convergence requirement for the discretized form of (7.1), which must be used in computers this turns out to be AT/(Ap)2normalized forms of r and t. Often, Ar/(Ap)2 = 1/6 is used to ensure better convergence. Of course, any procedure requires a reaction scheme, values of diffusion and rate coefficients, and a statement about initial number of species and their distribution in space (vide infra). 

Given ng and 1(R, p), Eqs. (7.32a, b) can be integrated successively from r = R to a large value of r. By definition Z(R, p) = -y where yis the recombination probability in presence of scavenger. Only for the correct value of ydo the solutions of (7.32a, b) smoothly vanish asymptotically as r—-o° otherwise, they diverge. Thus, the mathematics is reduced to a numerical eigenvalue problem of finding the correct value of I(R, p). 

Cycled Feed. The qualitative interpretation of responses to steps and pulses is often possible, but the quantitative exploitation of the data requires the numerical integration of nonlinear differential equations incorporated into a program for the search for the best parameters. A sinusoidal variation of a feed component concentration around a steady state value can be analyzed by the well developed methods of linear analysis if the relative amplitudes of the responses are under about 0.1. The application of these ideas to a modulated molecular beam was developed by Jones et al. ( 7) in 1972. A number of simple sequences of linear steps produces frequency responses shown in Fig. 7 (7). Here e is the ratio of product to reactant amplitude, n is the sticking probability, w is the forcing frequency, and k is the desorption rate constant for the product. For the series process k- is the rate constant of the surface reaction, and for the branched process P is the fraction reacting through path 1 and desorbing with a rate constant k. This method has recently been applied to the decomposition of hydrazine on Ir(lll) by Merrill and Sawin (35). 

TTie value of R and characteristics of CSD are easily obtained by using the numerical table of probability integral of %2-distribution (2). 

In an analogous fashion the numerically obtained transition probabilities shown in Fig. 14.6 can be converted to cross sections by integrating over impact parameter and averaging over the possible collision velocities. Collisions with v 1 E do not have a single allowed value of 6. However, since the lineshape of Fig. 14.6(b) is simple, some averaging over the possible values of 6 has no appreciable effect. For v E, 6 and 0 are fixed, at 90° and 0°, so the integration over impact parameter is only over b, and the calculated v E cross section is very similar to Fig. 14.6. 

The parameter A determines the probability distribution F(< >) and, consequently, the normalization constant Cp according to equation 15. This equation was numerically integrated for a range of A values. The calculated values of Cy are shown as a function of the parameter A in Figure 10. Compared with low values of Cp high values of Cy mean that the molecular beam is proportionally more intense near the nozzle center line. 

Note that the deviation of the transformed population from 100% should not be understood as the error in our approximation (5.8). The integrated total transition probability according to (5.8) is V = 0.879, which is in excellent agreement with the value V = 0.864 obtained by numerical solution of (1.2). This deviation from 100% efficiency is intrinsic, i.e., it derives from the spread of the wavepacket. The total transition probability can be improved by increasing the laser intensity. This is because the range of A increases, as can be seen from (5.28). For instance, if we use I = 4.0TW/cm2, then the total transition probability reaches 93 94%. Since we have to be careful about multiphon processes, it is better not to use very high intensities. 

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