Evaluating the Accuracy of Chebyshev’s Inequality for Probability Calculation: A Simulation Study

Abstract

This paper aims to evaluate the accuracy of probability calculation using Chebyshev’s inequality based on simulation study. We consider symmetric (Normal (3,1.5² ), Laplace (3, 2  ) Beta (7.7 ) t₅) positively skewed, negatively skewed (5 χ2,  Beta (3, 8  ) Gamma (5,1)), (Beta (7, 2)), Exponential (5) and Uniform (0, 1 ) distributions, fx(x) in our simulation study to measure the performance of Chebyshev’s inequality. We then calculate Pr (μ − kσ ≤ X ≤ μ + kσ ) for ~ ( ) X X f x , μ = E ( X ) and σ 2 =Var ( X ), and compare this with the approximated probability obtained from Chebyshev’s inequality to measure the accuracy of Chebyshev’s inequality. From our simulation study, it is observed that loss due to using Chebyshev’s inequality for probability calculation is the least and the maximum when fx(x) is the Exponential and the Beta (symmetric) distributions, respectively for k ≥ 2.5. Moreover, the value of Pr (μ − kσ ≤ X ≤ μ + kσ ) calculated using Chebyshev’s inequality is underapproximated for all the probability distributions considered in the study.

Dhaka Univ. J. Sci. 71(1): 76-81, 2023 (Jan)