Rare event approximation

This problem is usually solved by neglecting small quantities of higher order which result from products of minimal cut sets. This is known as the rare event approximation, which is often applicable because failure probabilities tend to be small. The expected value of the structure function is then approximated by... 

Simply summing the OR s events is a convenient approximation that only holds when all inputs are small (of the order of lE-3 or smaller). This is known as rare event approximation , see NASA Fault Tree Handbook paragraph 7.1 for more information. 

A final remark should be made regarding the use of the minimal cut set approximation (rare event approximation) method in determining the top event probability (chance). Using this method, the obtained top event probabihty (chance) is an upper bound. The question then arises whether this approximation introduces an element of uncertainty that has a larger... 

As the risk assessment progresses, you will see that quite a number of event trees are generated. To help alleviate this overabundance of data, it becomes useful to truncate some of the less important information. If the event tree has very small risk expectation values, such as the last two branches on the tree in Figure 14.1, then the tree can be pruned or dropped. Rare-event approximation can be used to further reduce the number of branches of the event tree. 

Once the FuBI of minimal cut-sets have been determined, the TE s FuBI (i.e. Njf) is calculated by using the rare events approximation formula ... 

If the form of the distributional model is only approximately eorreet, then the tails may differ substantially from the tails of the aetual distribution. This is beeause the model parameters, related to low order moments, are determined from typieal rather than rare events. In this ease, design deeisions will be satisfaetory for bulk... 

In 1990 approximately 50 cases of anterior uveitis were reported in the United Kingdom, where metipranolol had been available since 1986. At one hospital the incidence of uveitis in patients using 0.6% metipranolol was 14% (15 of 109). All cases resolved with appropriate management. Drug-induced anterior uveitis has also been reported as a rare event in the United States, but a true causal relationship has not been definitely established. 

Polymer Blends. Let us now apply this technique to the study of polymer blends. We will assume that the two materials have different sets of (mechanical) transition temperatures and that energy spectra of the luminescent light given off by the two polymers are distinguishably different. If the two polymers are completely miscible on a molecular scale (a rare event indeed), we can expect that many of the luminescence maxima will be seen at temperatures different from those of the blend components. This will certainly apply to the Tg of the miscible blend which can be approximated by ... 

The fragmentation of an alkane can be understood by assuming that a C—C bond is broken, sometimes accompanied by a hydrogen shift in order to minimize the energy requirements this is, to a first approximation, a rather satisfactory model. In the case of a branched alkane the bond to a tertiary or quaternary carbon atom is more easily broken, a fact that allows the determination of the branching point(s) with a good level of confidence. Except for small alkanes, the cleavage of C—H bonds seems to be a rare event. 

To the knowledge of the author, no analytical solution (neither exact nor approximate) exists for finding the design point of the problem posed in Equation 14. Hence, it is necessary to use an optimization technique for determining x. Numerical validation has shown that a technique which is convenient fur such purpose is the so-called Cross Entropy (CE) optimization method (see, e.g. (Rubinstein 1999)). This method is a gradient-free algorithm and, hence, is suitable for dealing with structural non linearities. It has been applied to a number of optimization problems in different fields and it can be used for rare event simulation as well. For specific details about its theoretical basis and its features, it is referred to the literature (see, e.g. (de Boer et al. 2005) for a comprehensive tutorial). 

The opportunity to test fire protection systems under real situations is, thankfulfy, a rare event in most organizations. Consequently, risk determination is difficult and the best assessment can only be an approximation of future outcomes. Nevertheless, assessments and approximations are critical factors in setting standards and measuring performance. To position a fire extinguisher on a wall and expect some untrained employee to use the device is unacceptable. To have the extinguisher, maintain it, and not expect it to be used to protect life and property is nonsense. The well conducted safety audit will discover the strengths and weaknesses in the system and provide a blueprint for any action plans necessary for improvement. 

Within emergency medicine, it is a common perception that chemical incidents are rare events. It has often been quoted in the past that there are approximately 1000 acute chemical incidents per year in England and Wales that may impact human health and that the majority of these events involve less than four casualties. For the purposes of public health, a chemical incident is defined as an event in which there is, or could be, exposure of the public to chemical substances that cause, or have the potential to cause, ill health . Hospital staff and emergency services personnel are considered to be members of the public in this definition. In 2005, the UK Health Protection Agency recorded more than 1000 chemical incidents, and it was estimated that approximately 27,000 people may have been exposed, of whom about 3000 reported with symptoms. It is thought that the actoal numbers may be considerably higher. 

When the value of p is very close to zero in Eq. (9-77), so that the occurrence of the event is rare, the binomial distribution can be approximated by the Poisson distribution with X = np when n > 50 while npi < 5. 

For short times, the correlation function (7(f) depends on the microscopic details of the dynamics as the system crosses from to 38. These motions take place on a molecular time scale rmoi essentially equal to the time required to move through the transition region. For times f larger than rmoi but still very small compared to the reaction time rrxn (if the crossing event is rare rrxn L> rmoi such that such an intermediate time regime exists), C(f) can be replaced by an approximation linear in time. Using the detailed balance condition k jk = h )/ h ) [33] one then obtains... 

Approximately 10% of new chemical entities (NCEs) show serious adverse drug reactions (ADRs) after market launch. Such events usually result in new black box warnings by the US Food and Drug Administration (FDA), label change or market withdrawal. The most common causes for these actions are hepatic toxicity, hematologic toxicity and cardiovascular toxicity [2], Reasons for such ADRs, which are identified only after NCEs are launched on the market, include the narrow spectrum of clinical disorders and participating patient profiles in clinical studies as well as the fact that serious ADRs are often rare and that the number of patient exposures required to identify such occurrences sometimes may range over a few millions [3],... 

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