We investigated the variation comparison between the $ F $-distribution and the normal distribution. Let $ X_{d_1, d_2} $ be an $ F $-random variable with numerator and denominator degrees of freedom as $ d_1 $ and $ d_2 $, respectively. We proved that, for $ d_1\in\{1, 2, 3, 4\} $ and $ 5\le d_2\in\mathbb{N} $, $ P\{|X_{d_1, d_2}-{\rm E}[X_{d_1, d_2}]|\le \sqrt{{\rm Var}(X_{d_1, d_2})}\} > P\{|Z|\leq1\} $, where $ Z $ is a standard normal random variable. Moreover, for each fixed $ d_1\in\{1, 2, 3, 4\} $, $ \inf_{d_2\ge5}P\{|X_{d_1, d_2}-{\rm E}[X_{d_1, d_2}]|\le \sqrt{{\rm Var}(X_{d_1, d_2})}\} = P\{|\chi^2_{d_1}-{\rm E}[\chi^2_{d_1}]|\le \sqrt{{\rm Var}(\chi^2_{d_1})}\} $, where $ \chi^2_{d_1} $ is a $ \chi^2(d_1) $-random variable. These four structurally representative cases capture the main features of the problem, while the case $ d_1\geq5 $ remains open.
Citation: Ping Sun, Ze-Chun Hu, Wei Sun. Variation comparison between the $ F $-distribution and the normal distribution[J]. AIMS Mathematics, 2026, 11(10): 33452-33484. doi: 10.3934/math.20261309
We investigated the variation comparison between the $ F $-distribution and the normal distribution. Let $ X_{d_1, d_2} $ be an $ F $-random variable with numerator and denominator degrees of freedom as $ d_1 $ and $ d_2 $, respectively. We proved that, for $ d_1\in\{1, 2, 3, 4\} $ and $ 5\le d_2\in\mathbb{N} $, $ P\{|X_{d_1, d_2}-{\rm E}[X_{d_1, d_2}]|\le \sqrt{{\rm Var}(X_{d_1, d_2})}\} > P\{|Z|\leq1\} $, where $ Z $ is a standard normal random variable. Moreover, for each fixed $ d_1\in\{1, 2, 3, 4\} $, $ \inf_{d_2\ge5}P\{|X_{d_1, d_2}-{\rm E}[X_{d_1, d_2}]|\le \sqrt{{\rm Var}(X_{d_1, d_2})}\} = P\{|\chi^2_{d_1}-{\rm E}[\chi^2_{d_1}]|\le \sqrt{{\rm Var}(\chi^2_{d_1})}\} $, where $ \chi^2_{d_1} $ is a $ \chi^2(d_1) $-random variable. These four structurally representative cases capture the main features of the problem, while the case $ d_1\geq5 $ remains open.
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Ping Sun, Ze-Chun Hu, Wei Sun. Variation comparison between the $ F $-distribution and the normal distribution[J]. AIMS Mathematics, 2026, 11(10): 33452-33484. doi: 10.3934/math.20261309
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