Probability theorems

After defining fundamental terms used in probability and introducing set notation for events, we consider probability theorems facilitating tlie calculation of the probabilities of complex events. Conditional probability and tlie concept of independence lead to Bayes theorem and tlie means it provides for revision of probabilities on tlie basis of additional evidence. Random variables, llicir probability distributions, and expected values provide tlie means... 

Consider the irreversible two-compartment model with survival, distribution, and density functions starting time, the molecules are present only in the first compartment. The state probability p (t) that a molecule is in compartment 1 at time t is state probability p2 (/,) that a molecule survives in compartment 2 after time t depends on the length of the time interval a between entry and the 1 to 2 transition, and the interval I, a between this event and departure from the system. To evaluate this probability, consider the partition 0 = ai < a.2 < < o.n 1 < an = t and the n — 1 mutually exclusive events that the molecule leaves the compartment 1 between the time instants a, i and a,. By applying the total probability theorem (cf. Appendix D), p2 (t) is expressed as... 

Total probability theorem. Given n mutually exclusive events A, ..., An, whose probabilities sum to unity, then... 

According to the total probability theorem, integrating Lk w)s k ) dw gives the probability Rk that the sum of the losses from k loss-generating bets will be smaller than the sum of the benefits from n — k successful bets ... 

In order to evaluate the accuracy of this approach, the method has been applied to a simple random vibration problem, namely the estimation of the spectrum standard deviation of the displacement of a SDOF linear system with damping ratio = 0.05. To this end use is made of the total probability theorem, applied to estimate the unconditional variance of the structural displacement d 6), considered as a function of the ground motion parameters 6 = cwg, Vg, Ag, as follows ... 

This follows from the total probability theorem. See, for example, pp 64 of Gnedenko (1964). 

The PDF of the experimental data p(d 0) can be interpreted as a measure of how good a model succeeds in explaining the observatiOTis d As this PDF reflects the likelihood of observing the data d when the model is parameterized by 0, it is also referred to as the likelihood function E(0 d). Since the data set d is fixed, this function in fact no longer represents a cmiditional PDF and can be denoted as L(0 d) in the following, however, the common notation of E(0 d) is pertained. The likelihood function is determined according to the total probability theorem in terms of the probabilistic models of the measurement and modeling errors ... 

Based on total probability theorem, Eq. 1 allows deconstructing the problem in four steps (i) hazard analysis, (ii) structural analysis, (iii) damage analysis, and (iv) loss analysis. Each step carries out a specific generalized variable intensity measure (JM), engineering demand parameter (EDP), damage measure DM), and decision variable DV). The key issue of PBEE methodology is to identify and quantify DV of primary interest to the decision makers with consideration to all important uncertainties. DPs have been defined in terms of different quantities, such as repair costs, downtime, and casualty rates. 

Sample Distributions and the Central Limit Theorem Let s return to the problem of determining a penny s mass to explore the relationship between a population s distribution and the distribution of samples drawn from that population. The data shown in Tables 4.1 and 4.10 are insufficient for our purpose because they are not large enough to give a useful picture of their respective probability distributions. A better picture of the probability distribution requires a larger sample, such as that shown in Table 4.12, for which X is 3.095 and is 0.0012. 

J Cornfield. In DL Meyer, RO Collier, eds. The Frequency Theory of Probability, Bayes Theorem, and Sequential Clinical Trials. Bloomington, In Phi Delta Kappa, 1970, pp 1-28. 

Vj attdHj, or r, and andP andV andli. Upon comparison to go from failure to success one or vice versa probability is complimented and logic is reversed AND goes OR and vice versa, as indicated by de Morgan s theorem (Table 2.2-1). 

Theorem 1 says that tlie probability that A does not occur is one minus tlie probability tliat A occurs. Thcorcni 2 siiys that the probability of any event lies between 0 and 1. Theorem 3, tlie addition tlieorein, provides an alternative way of calculating tlie probability of tlie union of two events as tlie sum of tlieir... 

As a example of the application of Bayes theorem, suppose tliat 50% of a company s manufactured output comes from a New York plant, 30% from a Permsylvania plant, and 20% from a Delaware plant. On die basis of plant records it is estimated diat defective items constitute 1% of the output of the New York plant, 3% of the Pennsylvania plant, and 4% of die Delaware plant. If an item selected at random from die company s manufactured output is found to be defective, what are die revised probabilities diat die item was produced, by each of die diree plants ... 

For anotlier example of the use of Bayes theorem, suppose that the probability is 0.80 tliat an airplane crash due to structural failure is diagnosed correctly. Suppose, in addition, tliat tlie probability is 0.30 tliat an airplane crash not due to structural failure is incorrectly attributed to structural failure. If 35% of all airplane crashes are due to structural failure, wliat is tlie probability that an airplane crash was due to structural failure, given tliat it lias been so diagnosed Let Ai be tlie event tliat structural failure is tlie cause of tlie airplane crash. Let A2 be tlie event tliat tlie cause is otlier tlian structural failure. Let B be tlie event tliat tlie airplane crash is diagnosed as being due to structural failure. Tlien... 

Bayes theorem provides tlie mechanism for revising prior probabilities, i.e., for converting tliem into posterior probabilities on tlie basis of the observed occurrence of some given event. ... 

The one-to-one correspondence of alloy and host sites is seen explicitly in Eq. (4). For the moment we now concentrate on the transition probability This quantity is proportional to the density of impurities and, according to the optical theorem, is given by... 

If a well proves productive, the ensuing completion operation may require an area in excess of the drilling area. This may mean allocations for frac tank placement, blenders, pump trucks, bulk trucks and nitrogen trucks. In today s economic climate, the operator should weigh the probability of success, Bayes theorem (Equation 4-373), with the cost of constructing and reclaiming an additional area needed for stimulation (Equation 4-374). Plans such as these... 

The Kolmogorov consistency theorem [gnto88] asserts that any set of self- and mutually- consistent probability functions Pj, j = 1,2,..., jV may be extended to a unique shift-invariant measure on F,... 

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