Total probability theorem

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 Since the data set d is fixed, this function in fact no longer represents a cmiditional PDF and can be denoted as L(0 d) in the following, however, the common notation of E(0 d) is pertained.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. 

This is the BINOMIAL THEOREM. Using a Taylor Expansion, we can find the total probability for items taken r at a time as ... 

This is an obvious adaptation of a theorem given by Feller. The chain takes an average of n/3 steps in the x direction, and the total probability of a walk from (0,, ) to (x>2 ) is the usual Gaussian. A similar modification of another theorem, which is valid for n+l n, is... 

It is an interesting Tact that just as the single s orbital is spherically symmetric, the summation or electron density of a set or three p orbitals, five d orbitals, or seven f orbitals is also spherical (UnsBld s theorem). Thus, although it might appear as though an atom such as neon with a filled set of sand p orbitals would have a lumpy electron cloud, the total probability distribution is perfectly spherical... 

Let us consider the seismic reliability assessment of a structure and assume that the site seismic hazard is described by the earthquake magnitude M and the source-to-site distance R. The probability of structural failure P(F), that is, the probability of exceeding a threshold response level, can be expressed by the Theorem of Total Probability as ... 

Law of Total Probability n Also known as the Theorem on Total Probability or the Law of Alternatives. It states that the probability, P(A) of an event, A, is equal to the sum of the conditional probabilities of A, given events, Ej, P(A 1 ), times the probability of event, , for i = 1, 2, 3,. ..,N where Nisa. positive integer or infinity, and where Efi are non-overlapping and form a partition of a sample space that covers the sample space of A. This can be expressed as ... 

Theorem on Total Probability n An alternate name for law of total probability. 

Seismic hazard is quantified many ways. One is through a hazard curve, commonly depicted on an x-y chart where the x-axis measures shaking intensity at a site and the y-axis measures either exceedance probability in a specified period of time or exceedance rate in events per unit time. See Fig. 3 for an example Cornell (1968) applied the theorem of total probability to create a hazard curve. What follows here is a summary of current procedures to perform probabilistic seismic hazard analysis (PSHA), but is conceptually identical to Cornell s work. 

To estimate seismic hazard, one applies the theorem of total probability to combine the uncertain shaking at the site caused by a particular fault rupture and the occurrence frequency or probability of that rupture. Earth scientists create models called earthquake rupture forecasts that specify the locations and rates at which various fault produce earthquakes of various sizes, e.g., the Uniform California Earthquake Rupture Forecast version 2 (UCERF2, Field et al. 2007). The uncertain shaking given a fault rupture is quantified using a relationship variously called an attenuation relationship or a ground-motion prediction equation, such as the next-generation attenuation (NGA) relationships presented in the... 

One can now estimate the hazard curve by applying the theorem of total probability. Suppose otic always knew soil conditions V with certainty. Then... 

Equation 9 is known as Bayes theorem or Bayes rule. Using the theorem of total probability given by... 

In its extreme form the ergodic hypothesis is clearly untenable. Only probability statements can be made in statistical mechanics and these have nothing to do with sequences in time [117]. Not surprisingly, a totally convincing proof of the ergodic theorem in its many guises has not been formulated. The current concensus still is that an axiomatic basis, completely independent of ergodic theory should be chosen [115] for the construction of statistical mechanics. 

Equation (21) already has the form of a fluctuation theorem. However, in order to get a proper flucmation theorem we need to specify relations between probabilities for physically measurable observables rather than paths. From Eq. (21) it is straightforward to derive a fluctuation theorem for the total dissipation S. Let us take b C) = With this choice we get... 

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