The Bienaymé-Chebyshev Inequality provides a quantitative bound on the level of confidence of a measurement with known combined standard uncertainty and assumed coverage factor. The result is independent of the detailed nature of the probability distribution that characterizes knowledge of the measurand.

In this paper, by the Chebyshev-type inequalities we define three mappings, investigate their main properties, give some refinements for Chebyshev-type inequalities, obtain some applications.

In this paper, we introduce the generalized left-side and right-side fractional integral operators with a certain modified ML kernel. We investigate Chebyshev inequality via general family of operators. Moreover, derive new results type inequalities for finite products functions. addition, establish an estimate functional by using From our above-mentioned results, find similar some specialized ...

1 Concentration Inequalities Lemma 1 (Markov’s inequality). Let X be a non-negative random variable. For all λ > 0, Pr[X > λ] ≤ E[X] λ Lemma 2 (Chebyshev Inequality). For all λ > 0, Pr[|X −E[X]| > λ] ≤ var[X] λ2 Lemma 3 (The Chernoff Bound: Upper bound). Let X1, X2, ..., Xn be independent random variables taking values in {0, 1} with E[Xi] = pi. Let X = ∑n i=1Xi, and μ = E[X]. Then the followin...

According to a result of S. N. Bernstein, «-point Chebyshev quadrature formulas, with all nodes real, do not exist when n = 8 or n ä 10. Modifications of such quadrature formulas have recently been suggested by R. E. Barnhill, J. E. Dennis, Jr. and G. M. Nielson, and by D. Kahaner. We establish here certain empirical observations made by these authors, notably the presence of multiple nodes. We...

In this paper, we propose a method to detect anomalous behaviour using heterogenous data. This method detects anomalies based on the recently introduced approach known as Recursive Density Estimation (RDE) and the so called eccentricity. This method does not require prior assumptions to be made on the type of the data distribution. A simplified form of the well-known Chebyshev condition (inequa...

This paper is concerned with the parameter estimation problem for a class of diffusion process from discrete observations. The approximate likelihood function given by using Riemann sum and an Itˆo to integrals in continuous-time function. consistency maximum estimator asymptotic normality error are proved applying martingale moment inequality, Holder’s Chebyshev B-D-G inequality uniform ergodi...