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Mathematical Preliminaries
Published in Craig Friedman, Sven Sandow, Utility-Based Learning from Data, 2016
(a) Use the Markov inequality to prove the Chebyshev inequality(b) Using the Schwarz inequality, show that − 1 ≤ ρ [X, Y] ≤ 1
Properties of Algebraic Inequalities
Published in Michael T. Todinov, Risk and Uncertainty Reduction by Using Algebraic Inequalities, 2020
Chebyshev’s inequality also provides a useful tight upper bound of the values of a random variable for values of the constant k greater than the standard deviation σ.
Discrete Random Variables and Their Distributions
Published in Michael Baron, Probability and Statistics for Computer Scientists, 2019
Chebyshev’s inequality shows that in general, higher variance implies higher probabilities of large deviations, and this increases the riskfor a random variable to take values far from its expectation.
Differentially private containment control for multi-agent systems
Published in International Journal of Systems Science, 2022
Zewei Yang, Yurong Liu, Wenbing Zhang, Fawaz E. Alsaadi, Khalid H. Alharbi
In particular, results available in the literature mainly focus on consensus problems, and there are few results on containment control problems due to the difficulties in estimating the global convergence accuracy in the presence of multiple leaders. When considering differentially private consensus problems, states of agents among the MAS converge to the average initial states in expectation. Chebyshev inequality emphasises the degree that the values of random variables are concentrated around their expectation. Therefore, it is natural adopting Chebyshev inequality to estimate the global probability that agreement value falls into the neighbourhood of the average initial states. Compared with the consensus problems (Gao et al., 2019; Huang et al., 2012; Nozari et al., 2015, 2017; A. Wang et al., 2019), when considering containment control problems, followers' states falling into the convex hull formed by the leaders' states is the ultimate goal, thus the convex combination of leaders' states for each follower may be different in containment control behaviours. In dealing with the global estimation of the probability that each follower's state falls into the neighbourhood of that convex hull formed by multiple leaders, Chebyshev inequality is no longer applicable since there may be no agreement value in containment control problems. A new method is urgent to utilised to estimate the overall convergence accuracy. To sum up, it is of great importance to study the privacy protection in containment control of the MASs.
Combining predetermined and measured assembly time techniques: Parameter estimation, regression and case study of fenestration industry
Published in International Journal of Production Research, 2019
Vladimir Polotski, Yvan Beauregard, Arthur Franzoni
It is well known that the assumption about the normal distribution of the measurements are sometimes difficult to verify thus alternative approaches are worth exploring. The technique based on the Chebyshev inequality that does not require any assumption about the distribution of the measurements. Chebyshev inequality asserts for a random variable X with mean μ and variance σ (both finite) that the probability of X deviating from μ by more than is less than : Applying Chebyshev inequality to the statistics of the sample average the possible reasoning is as follows. Keeping the notation introduced by formulas (1), let us consider two random variables – the sample means and . They are the average values of and , respectively, which in turn are the realisations of random variables and , with the (population) means and and variances and . Therefore the means of are and their variances (for uncorrelated realisations - that holds in our case) are and , respectively (Seltman 2015). Final step consists of considering the sum of two sample averages . As its mean and variance are and , the Chebyshev inequality asserts with probability (confidence) that population mean belongs to an interval with Comparing this result with expressions (7) it is important to emphasise that the interval obtained through Chebyshev technique has the same order over the variance (proportional to ) as (7), but with larger factor. For example 95% results in the factor value of 4.47 in (9) comparing to factor value of in (7) (for sample size larger then 10).