{"title":"评价切比雪夫不等式在概率计算中的准确性:模拟研究","authors":"Tasmiah Afrin Emma, A. Sajib, Sabina Sharmin","doi":"10.3329/dujs.v71i1.65276","DOIUrl":null,"url":null,"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.\nDhaka Univ. J. Sci. 71(1): 76-81, 2023 (Jan)","PeriodicalId":11280,"journal":{"name":"Dhaka University Journal of Science","volume":"157 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Evaluating the Accuracy of Chebyshev’s Inequality for Probability Calculation: A Simulation Study\",\"authors\":\"Tasmiah Afrin Emma, A. Sajib, Sabina Sharmin\",\"doi\":\"10.3329/dujs.v71i1.65276\",\"DOIUrl\":null,\"url\":null,\"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.\\nDhaka Univ. J. Sci. 71(1): 76-81, 2023 (Jan)\",\"PeriodicalId\":11280,\"journal\":{\"name\":\"Dhaka University Journal of Science\",\"volume\":\"157 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-05-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Dhaka University Journal of Science\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3329/dujs.v71i1.65276\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Dhaka University Journal of Science","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3329/dujs.v71i1.65276","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
本文旨在通过仿真研究,评价利用切比雪夫不等式进行概率计算的准确性。在我们的模拟研究中,我们考虑对称(Normal(3,1.52),拉普拉斯(3,2)Beta (7.7) t5)正偏态,负偏态(5 χ2, Beta (3,8) Gamma (5,1)), (Beta(7,2)),指数(5)和均匀(0,1)分布,fx(x)来衡量Chebyshev不等式的性能。然后我们计算了~ ()X X f X, μ = E (X)和σ 2 =Var (X)的Pr (μ−kσ≤X≤μ + kσ),并将其与由Chebyshev不等式得到的近似概率进行比较,以衡量Chebyshev不等式的精度。从我们的模拟研究中可以观察到,当k≥2.5时,fx(x)为指数分布和Beta(对称)分布时,使用Chebyshev不等式进行概率计算的损失分别最小和最大。此外,用Chebyshev不等式计算的Pr (μ−kσ≤X≤μ + kσ)的值对于研究中考虑的所有概率分布都是过近似值。达卡大学学报(自然科学版),71(1):76- 81,2023 (1)
Evaluating the Accuracy of Chebyshev’s Inequality for Probability Calculation: A Simulation Study
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)