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Basic Concepts in Probability
Published in X. Rong Li, Probability, Random Signals, and Statistics, 2017
Total probability theorem is often useful for the calculation of the unconditional probability of an event P{B} knowing various conditional probabilities of the events P{B|Ai} and the probabilities of the conditioning events P{Ai}. Intuitively, it provides a way to find an “effect” from its “causes”: It calculates the probability of an “effect” (event B) from the probabilities of all its possible “causes” (events Ai’s) and the relationships between these possible “causes” and “effect” (P{B|Ai}).
A new robust dynamic Bayesian network approach for disruption risk assessment under the supply chain ripple effect
Published in International Journal of Production Research, 2021
Ming Liu, Zhongzheng Liu, Feng Chu, Feifeng Zheng, Chengbin Chu
According to the DBN introduced by Hosseini, Ivanov, and Dolgui (2019b), the transition of the supplier i's state from in the previous time period t−1 into in the present time period t can be described by a probability , where , . Namely, , which is a conditional probability, conditioning on only one variable. All state transition relationships for the supplier i can be represented by a Markov transition matrix as follows: where is an () square matrix, describing a conditional probability distribution. For the manufacturer, however, its state in the present time period t depends on not only its state in the previous time period t−1, but also all the suppliers' states in the present time period t. Thus, the state transition relationships for the manufacturer cannot be simply described by a Markov transition matrix. Instead, a conditional probability table (CPT) is utilised for this purpose, which also describes a conditional probability distribution, conditioning on multiple variables.
Designing variable sampling plans based on lifetime performance index under failure censoring reliability tests
Published in Quality Engineering, 2020
Hasan Rasay, Farnoosh Naderkhani, Amir Mohammad Golmohammadi
For a given value of lifetime non-conforming rate δ, under the RGS plan, the acceptance probability of the lot denoted by can be computed as follows: such that can be calculated using Eq. (14) and is given as follows: Please note that is derived using the law of total probability and by conditioning on the event that acceptance of the lot happens at the sampling (). This can be stated as follows: This completes the calculation of the OC curve. Next, the optimization problem of the RGS plan will be discussed.
From Least Squares to Signal Processing and Particle Filtering
Published in Technometrics, 2018
Nozer D. Singpurwalla, Nicholas G. Polson, Refik Soyer
Incorporating the factorization of (7.3) into the relationship given before, we have where by law of total probability, by conditioning on θt + 1. Assuming θt + 1 is independent of Yt given θt, and Yt + 1 is independent of θt and Yt given θt + 1, we have Thus, (7.4) simplifies as Equation (7.5) parallels (7.2) and is an alternate to it. It encapsulates the “update first-propagate next” protocol. Note that the key difference between (7.2) and (7.5) pertains to the feature that the former entailed P(Yt + 1|θt + 1) whereas the latter entails P(Yt + 1|θt). With Yt + 1 observed as yt + 1, P(Yt + 1|θt + 1) spawns the filtering likelihood whereas P(Yt + 1|θt) spawns the smoothing likelihood. An advantage of the smoothing likelihood over the filtering likelihood is that were yt an outlier but yt + 1 not, then a consideration of a likelihood based on yt + 1 would diminish the ill effects of yt.