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Introduction to Logic and Probability
Published in Sriraman Sridharan, R. Balakrishnan, Foundations of Discrete Mathematics with Algorithms and Programming, 2019
Sriraman Sridharan, R. Balakrishnan
Consider a finite sample space consisting of n points. If the probability of each point is 1 / n, then S is an equiprobable space. If an event E⊂S $ E\subset S $ consists of k elements, then p(E)=k/n $ p(E)=k/n $ . Hence, P(E)=number of elements in Enumber of elements in S $$ P(E)=\frac{\text{ number} \text{ of} \text{ elements} \text{ in} \text{ E}}{\text{ number} \text{ of} \text{ elements} \text{ in} \text{ S}} $$
Introduction to Digital Modulation
Published in Sibley Martin, Modern Telecommunications, 2018
As can be seen from Figure 3.7, the threshold voltage, VT, for the comparator is set midway between logic 0 and logic 1. This assumes that there is an equal probability of a 1 or a 0 – they are equiprobable. This is a reasonable assumption because most transmission links are designed to have equiprobable symbols to avoid effects such as baseline wander. The next point to note is that every voltage above VT is registered as logic 1 and everything below VT is registered as logic 0. The Gaussian distributions on 0 and V represent the noise in the system. The two areas of interest are to the left and right of VT. The area to the right represents a logic 1 being generated by noise on the logic 0 causing a false threshold crossing. Similarly, the area to the left of VT is noise corrupting a pulse (logic 1) so that the threshold crossing is not registered. The crossover region is of interest because that is where the detection errors occur.
Basics of Probability
Published in Aliakbar Montazer Haghighi, Indika Wickramasinghe, Probability, Statistics, and Stochastic Processes for Engineers and Scientists, 2020
Aliakbar Montazer Haghighi, Indika Wickramasinghe
If a sample space is finite with n elements and all its outcomes have the same chance to occur, then the probability of any of the sample points occurring is assumed to be 1/n, and the sample space is referred to as equiprobable. The concept of equiprobability in case of an infinite sample space is referred to as the uniform measure. Naturally, in this case, the aforementioned assumption of probability is not possible. We will later address this idea. The notion of equiprobable measure is an example of a discrete probability measure. In such a case, the choice of an event is referred to as the selection at random.
Multi-criteria decision-making for collaborative COVID-19 surge management and inter-hospital patients’ transfer optimisation
Published in International Journal of Production Research, 2023
Imene Elhachfi Essoussi, Malek Masmoudi, M. Zied Babai
To deal with the uncertainty of the number of ill patients requesting ICU beds, several samples of generated scenarios are produced based on the baseline scenarios SC1, SC2, and SC3. To generate a sample of scenarios, Monte Carlo sampling method is applied, which relies on the probability distribution of the random variable . Recall that applying the Monte Carlo method produces equiprobable scenarios in each sample and thus providing an estimate of the scenario's probability of occurrence. However, the solvability of the SAA model is very dependent on the sample size, which means that the problem may be intractable for large sizes, even when powerful optimisation software is used. The sample size is calibrated by computing a statistical gap based on the framework of (Shapiro, Dentcheva, and Ruszczynski 2009). Each scenario consists in generating a random number of patients in each period (of the 18 weeks) using a normal distribution with a mean given by the number of patients requesting ICU beds in the base scenario and a standard deviation given by 3% of the mean number of patients in each period. The 3% is chosen according to a recommendation of the forecasting study conducted by Petropoulos, Makridakis, and Stylianou (2022), which shows a relative forecast error of around 3% for the same COVID contamination period.
Selective AnDE based on attributes ranking by Maximin Conditional Mutual Information (MMCMI)
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2023
Shenglei Chen, Xin Ma, Linyuan Liu, Limin Wang
To decide whether two comparing algorithms have the equal chances of win, a standard binomial sign test (Demšar, 2006) is applied to these records. Given the null hypothesis that wins and losses are equiprobable, the binomial test indicates the probability of observing the specified numbers of win and loss. In our analysis, the number of draws is divided equally to the number of wins and losses. If the resulted numbers are not integers, the number of wins will be rounded up to the next integer, while the number of losses will be rounded down to the nearest integer. We reject the hypothesis and consider the difference between the two algorithms significant if the value is less than the critical value 0.05, which is in bold font. The value we reported is the outcome of a two-tailed test. For example, value of ASAODE against AODE in terms of ZOL is 0.0011, which is the chance of observing either 49 or more wins, or 21 or fewer wins, in 70 comparisons. Since 0.0011 is less than 0.5, we can draw a conclusion that ASAODE is significantly better than AODE in terms of ZOL from the binomial sign test.
Distribution planning for multi-echelon networks considering multiple sourcing and lateral transshipments
Published in International Journal of Production Research, 2020
Mehdi Firoozi, M. Zied Babai, Walid Klibi, Yves Ducq
As mentioned, the two-stage stochastic multi-echelon inventory optimisation model (1)–(14) is intractable due to the inherent combinatorial complexity and the very large number of plausible scenarios necessary to shape entirely the demand process. The sample average approximation (SAA) technique (Shapiro 2007) is used to formulate an equivalent deterministic mixed-integer linear programme (MILP) that approximate the original stochastic model for a given sample of scenarios. The SAA method has been widely used in the recent years to find near-optimal solutions for stochastic problems in the supply chain (Klibi et al. 2010; Benyoucef, Xie, and Tanonkou 2013; Özdemir, Yücesan, and Herer 2013; Amiri-Aref, Klibi, and Babai 2018). To generate a sample of scenarios, Monte Carlo sampling method is applied, which rely on the probability distribution of the random variables. Applying Monte Carlo method, produces equiprobable scenarios in a given sample and thus provides an estimate of the scenario’s probability occurrence. Thus, a sample of independent demand scenarios is produced here, with , and transforms the original model (1)–(14) to the following SAA programme: subject to constraints (2)–(14) .