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Introduction to Random Signals
Published in Shaila Dinkar Apte, Random Signal Processing, 2017
The term event is defined. When the experiment is performed to generate some output, it is called as an event. The output of the experiment is called as the outcome of the experiment. Probability=limn→∞nHn is defined in terms of relative frequency. The conditional probability is given by P(A/B)=P(A∩B)P(B). Bayes’ theorem is introduced, which is the theorem of inverse probability.
A New Class of Mixture Probability Models with Applications
Published in American Journal of Mathematical and Management Sciences, 2019
Avishek Mallick, Indranil Ghosh
For each parameter combination, a random sample of appropriate size is simulated from the GTPW distribution by inverse probability integral transform and the parameters are estimated by the maximizing the log-likelihood function using the limited-memory quasi-Newton algorithm “L-BFGS” in R. This process is replicated 1000 times and the estimated biases and the estimated standard deviations are presented in Tables 1 and 2.
A sustainable road pricing oriented bilevel optimization approach under multiple environmental uncertainties
Published in International Journal of Sustainable Transportation, 2022
Ying Lv, Shanshan Wang, Ziyou Gao, Guanhui Cheng, Guohe Huang, Zhengbing He
Considering the uncertainties of the reality, the emission factors (g/km) on different links are presented by fuzzy triangular numbers. Furthermore, from the perspective of local pollution control, the pollutant emission will be regulated not to exceed the allowable amount on each link. Generally, the emission allowance will be decided by the local manager through conducting the effective data surveys among a certain amount of persons who have expertise in the field of traffic pollution control. Due to the complexities involved in the pollutant generation and diffusion processes, the respondents may show sticker or relaxed attitudes toward the regulation based on the estimations of emission pollution; thus, the emission allowance decided based on the survey data can be highly uncertain. In order to address the uncertainties, the investigator can ask the respondents to provide their suggestions about the allowable emission amounts at the low, medium and high levels, respectively. Accordingly, for each level, the collected data can be random associated with probability distributions. Therefore, the random-fuzzy variable can be used to specify the emission allowance under uncertainty, and it can be converted to the fuzzy number of at a desired pi risk level, where refers to the inverse probability distribution of the random-fuzzy variable. In detail, the related parameters used in the proposed BLP-SF model are presented in Table 1. In particular, the fuzzy emission factors can be specified as and the emission allowance can be presented as under a given risk level of 0.15 in the study.
Quality issue in forecasting problem of production and maintenance policy for production unit
Published in International Journal of Production Research, 2018
Zied Hajej, Nidhal Rezg, Ali Gharbi
The service-level constraint characterised by a probabilistic constraint is converted to an equivalent deterministic inequality as follows:where denotes the inverse probability distribution function of the inventory variable which depends on the service level β.