Gamma分布两个概率函数的极值
概率论
2023-03-31 v1
摘要
受Chvátal猜想与Tomaszewaki猜想的启发，我们研究Gamma分布两个概率函数的极值问题。设α , β \alpha,\beta α , β X α , β X_{\alpha,\beta} X α , β α \alpha α β \beta β P { X α , β ≤ E [ X α , β ] } P\{X_{\alpha,\beta}\le E[X_{\alpha,\beta}]\} P { X α , β ≤ E [ X α , β ]} P { ∣ X α , β − E [ X α , β ] ∣ ≤ V a r ( X α , β ) } P\{|X_{\alpha,\beta}-E[X_{\alpha,\beta}]|\le \sqrt{{\rm Var}(X_{\alpha,\beta})}\} P { ∣ X α , β − E [ X α , β ] ∣ ≤ Var ( X α , β ) } inf α , β P { X α , β ≤ E [ X α , β ] } = 1 2 \inf_{\alpha,\beta}P\{X_{\alpha,\beta}\le E[X_{\alpha,\beta}]\}=\frac{1}{2} inf α , β P { X α , β ≤ E [ X α , β ]} = 2 1 inf α , β P { ∣ X α , β − E [ X α , β ] ∣ ≤ V a r ( X α , β ) } = P { ∣ Z ∣ ≤ 1 } ≈ 0.6826 \inf_{\alpha,\beta}P\{|X_{\alpha,\beta}-E[X_{\alpha,\beta}]|\le \sqrt{{\rm Var}(X_{\alpha,\beta})}\}=P\{|Z|\le 1\}\approx 0.6826 inf α , β P { ∣ X α , β − E [ X α , β ] ∣ ≤ Var ( X α , β ) } = P { ∣ Z ∣ ≤ 1 } ≈ 0.6826 Z Z Z
引用
@article{arxiv.2303.17487,
title = {The extreme values of two probability functions for the Gamma distribution},
author = {Ping Sun and Ze-Chun Hu and Wei Sun},
journal= {arXiv preprint arXiv:2303.17487},
year = {2023}
}