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Creating a Reliable IIoT Framework to Prioritize Workplace Safety in Industries Involving Hazardous Processes
Published in Anand Sharma, Sunil Kumar Jangir, Manish Kumar, Dilip Kumar Choubey, Tarun Shrivastava, S. Balamurugan, Industrial Internet of Things, 2022
Nishant Sharma, Parveen Sultana H.
Equation (3.3) is the law of total probability. Stated in words, the law of total probability tells us that the probability of an event B occurring within a sample space partitioned into n, “Ai” parts is the sum total of the products of the probability that B could occur given that some event Ai has occurred times the probability that the event Ai could occur.
Mean and Insensitive. Random Permutations.
Published in Miklós Bóna, Combinatorics of Permutations, 2022
One basic and well-known application of conditional probabilities is Bayes' Theorem, which is also called the law of total probability. It can be found in any introductory probability textbook, such as [198]. It states the following.
A Stochastic Transport Model for the Cumulative Number of Fissions and Deposited Fission Energy
Published in Nuclear Science and Engineering, 2023
Anil K. Prinja, Patrick F. O’Rourke
In a Markov process, a knowledge of the state of a stochastic system at a given time is sufficient to determine what happens at the next step. In physical applications where the underlying stochastic process is Markovian, the goal is to obtain the probability of eventually reaching a particular state from the current state, expressed in the form of Chapman-Kolmogorov equations. In first step analysis, this is accomplished through the use of conditional probability and the Partition Theorem or the Law of Total Probability. We consider all possible first steps away from the current state and construct a probability balance of the contributions from each of these first step events that lead to the final state. Thus, let be the mutually exclusive first steps the process can take from current state. Then the probability that an event happens is given by
Stochastic analysis of dynamic stress amplification factors for slab track foundations
Published in International Journal of Rail Transportation, 2023
Hongwei Xie, Qiang Luo, Tengfei Wang, Liangwei Jiang, David P. Connolly
Assuming the cumulative probability of the track irregularity spectrum is at an operating speed of v, Equation (25) produces, corresponding to the cumulative probability p (, where denotes the random variable for DAFs, and denotes the random variable of ). The probability distribution of DAFs can be considered the marginal distribution of the joint distribution under the law of total probability. It can be derived from the conditional probability density function , expressed as:
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.