Theoretical probability

The appropriate theoretical probability distribution can be superimposed on the histogram ideally, scaling should be chosen to yield an area under the curve (integral from -°o to +o° equal to n w. (See program HISTO, option (ND).)... 

The distribution of a data set in the form of a histogram can always be plotted, without reference to theories and hypotheses. Once enough data have accumulated, there is the natural urge to see whether they fit an expected distribution function. To this end, both the experimental frequencies and the theoretical probabilities must be brought to a common scale a very eonvenient... 

Theoretical probabilities are determined using mathematical counting techniques and mathematical formulas. These probabilities are still just that predictions or likelihoods of events happening. 

Theoretical probability identifies the possible outcomes of a statistical experiment, and uses theoretical arguments to predict the probability of each. Many applications in chemistry take this form. In atomic and molecular structure problems, the general principles of quantum mechanics predict the probability functions. In other cases the theoretical predictions are based on assumptions about the chemical or physical behavior of a system. In all cases, the validity of these predictions must be tested by comparison with laboratory measurements of the behavior of the same random variable. A full determination of experimental probability, and the mean values that come from it, must be obtained and compared with the theoretical predictions. A theoretical prediction of probability can never be tested or interpreted with a single measurement. A large number of repeated measurements is necessary to reveal the true statistical behavior. 

Statistical models for the analysis of NMR data are used in two complementary approaches (Fig. 2) an analytical (model fitting) approach and a synthetic (computer simulation) approach. In the analytical approach, assigned NMR resonance intensities are fit to expected intensities based on statistical models. In the synthetic approach, spectral intensities are first calculated using reaction probabilities predicted by theoretical models these theoretical intensities are matched with those observed in the NMR spectrum. The calculation is based on theoretical probability expressions or Monte Carlo simulation. In an integrated approach, both methods are used for more complex systems. 

Fig. 9.1 The theoretical probability of molecular recognition based on a simple interaction model indicates that the likelihood of a unique binding mode decreases with increasing ligand complexity (A). The probability to experimentally detect a binding event is estimated to increase with complexity (B). The product probability for a so-called useful event, namely, the detection of a ligand with a unique binding mode, reaches a maximum at a medium ligand complexity (C). (See citation in text for details and discussion).

In theory, if we toss a die, we know that there is a 1 in 6 chance of getting a 2. But sometimes when we conduct dice-tossing experiments, we get statistics that do not reflect the theoretical probability. 

Figure 7. Bootstrap distribution for the mean 5-FU clearance data reported in Table 1 using unbalanced (top) and balanced bootstrapping (bottom) based on 1000 bootstrap replications. The solid line is the theoretical probability assuming a normal distribution. The unbalanced bootstrap distribution was normally distributed but the balanced bootstrap was slightly nonnormaL The arrow shows the observed mean clearance.

One may say, perhaps, that some factor must still be introduced in the theoretical expression to obtain the correct magnitude of 6 and the experimental observations offer a means of evaluating this. But, unfortunately, the theoretical probabilities do not have even the right relative values. They decrease with quantum number while for the experimental values Tolman and Badger found a decided increase. The absolute values which they calculated may be in error for the reasons given above, but more perfect resolution would be expected to increase the trend they observed rather than to eliminate it. It would seem, therefore, that the predictions of the new quantum theory, while they may apply to some ideal system, do not describe the conditions we have experimentally observed in the case of hydrogen chloride. 

Thus, the interval (3.03, 7.26) is a 0.95 (or 95%i) confidence interval for the effect A. It should be emphasized that the probability statement about the confidence level of 0.95 does not relate to the specific interval (3.03, 7.26) since this specific interval is an outcome of the specific sample used for the calculation, and it either contains the parameter A or it does not. It is a theoretical probability relating to a generic interval calculated from a sample following the steps we described above. Thus, if we could repeat the experiment many times, each time calculating a confidence interval in the way we have just done, we should expect 95%i of these intervals to contain the true mean effect A. Of course, when calculating a confidence interval from a sample, there is no way to tell whether the interval contains the parameter it is estimating or not. The confidence level provides us with a certain level of assurance that it is so, in the sense we have just described. 

Theoretical probability of tumor and toxic responses as a function of drug dose. The long-dashed Line represents the probability of tumor response and the short-dashed line represents the probability of a toxic response as a function of drug dose. For conventional chemotherapy, it is assumed that the therapeutic index (TI) is narrow and lies in proximity to the maximum tolerated dose (MTD) as defined by a tolerable level of toxicity. 

Theoretical probability of tumor response vs. the cumulative area under the concenlralion-vs.-time curve (cAUC). The therapeutic index (TI) is defined by the minimum effective regimen (MER) and maximum tolerated regimen (MTR) as determined by the cAUC. The MER is defined by the cAUC where a 95% probability exists of having one or more successes if the drug is at least 20% effective. 

The idea that covalent bonds may exist between adjacent metal ions is not new, but so far no good method has been available for defining the strength of such bonds. A possible definition of intercation covalehcy is to be found in the diminution of magnetic moment from that theoretically probable. For instance, suppose that trivalent nickel is found to have a moment of 2.7 instead of the theoretical 3.8. The degree of covalent bond formation may be expressed as the percentage 100 X (3.8 - 2.7)/3.8 = 29%. 

Figure 10,3 At point A there is a theoretical probability of one organism per unit. At point B there is a theoretical probability of organisms per unit, or one organism per 10 units...

If the temperature rises above another incompatibihty area may appear. Such an area has been observed in some polymer-solvent systems, which are characterized by a low critical solution temperature, LCST, T2c- In Figure 6.1, LCST is higher n UCST. The negative dependence of polymer-plasticizer compatibility on temperature suggests the theoretical probability that LCST also occurs in polymer-plasticizer systems such as PVC with either dii-sodecyl adipate, dibutyl phthalate, or tributyl phosphate. 

An alternative method is to compute the theoretical probability of an explosion event within the radius rp in each scenario the wind direction will move the explosive gas mixture to the plant. The advantage over the lee is that each scenario gives a contribution to the... 

The Fig 12. b) shows the cumulative theoretical probability for Weibull distribution. 

The Patterson method has now been largely replaced with a more powerful technique known as direct methods [32]. This is based on two fundamental physical principles. First, the electron density in the unit cell cannot be negative at any point, and so the large majority of possible sets of values for the phases of the various structure factors are not allowed. Secondly, the electron density in the cell is not randomly distributed, but is mainly concentrated in small volumes, which we identify as atoms. A consequence of these two principles is that certain theoretical probability relationships will exist between the phases of some sets of reflections (usually groups of three) that have particular combinations of Miller indices. It is therefore possible to assign probable phases to some reflections (usually the most intense ones), and then the positions of some or all of the heaviest atoms can be located. 

Only the last-mentioned situation is really dangerous. This means that only a very low percentage of faults (0.4 has not been detected by the self-monitoring unit and induced abnormal behaviour in the system. This result is to parallel with that of the theoretical probability of detecting faults by signature analysis which is of the order of 99.6 % in the present case. Even if these results are particular to a specific device, they are very promising for the future of signature analysis. 

Chi-square Goodness of Fit Test n The chi-square goodness of fit test is a type of chi-square test for quantifying how well a model predicts the observed data for a sample. The test uses the chi-square distribution to calculate the probability that the difference between the model and the observed data was due to chance alone. If the probability of having a difference that large is small, below a predetermined significance level, the model is rejected. The model is often a theoretical probability distribution. See Chi-Square Tests. 

ABSTRACT The new control system standard ISO 13849-1 deals with the theoretical probabilities of hypothetical individual events however, it avoids depicting them as relative frequencies. For the practical design engineers, a relative frequency approach is a more comprehensible form, because with the relative frequency a reconciliation with statistically acquired data is possible. This article closes some explanatory gaps caused by the one-sided emphasis on theoretical probability. In doing so, four contributions are provided in the context of field experience ... 

The one-sided polarisation of ISO 13849-1 towards theoretical probabilities is a disadvantage, because the relative frequencies would enable the theory to be objectively verified. For the practical design engineer, a reconciliation with statistically acquired data or experimental results is necessary, as well as with subjective empirical values, e.g. by enumerating the relevant events (in the numerator) with regard to the total quantity of all events (in the denominator). This fraction is the original definition of a probability (see e.g.. 

For practical reliability engineering, the relationship between an experimentally determined histogram and an empirical density is very important and it has been compiled in very easily comprehensible form e.g. in Montgomery several practical examples are adduced to explain how Bernoulli s Law of Large Numbers can be used to proceed from experimentally determined frequencies to theoretical probabilities and vice versa. In doing so, distribution functions can be derived, e.g. the normal distribution curve of Gauss. 

ABSTRACT As stated in part 1 of the article, standard ISO 13849-1 deals with the theoretical probabilities of hypothetical individual events and the possibiUty of recondUation of this theoretical approach with empiric field data is partly neglected. 

Fig. 3.33 Frequency distribution of MnS inclusions in carbon steels vs. maximum size as function of sulfur content (in particles n per cm ). Also show is the theoretical probability distribution...

Theoretical probability of ring opening to normal paraffins. 

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