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Title		Authors	PubMed ID
1	The infimum values of two probability functions for the Gamma distribution	Sun P; Hu ZC; Sun W;	38261930 MATHSTATS

Title:	The infimum values of two probability functions for the Gamma distribution
Authors:	Sun P, Hu ZC, Sun W
Link:	https://pubmed.ncbi.nlm.nih.gov/38261930/
DOI:	10.1186/s13660-024-03081-w
Publication:	Journal of inequalities and applications
Keywords:	Gamma distribution; Infimum value; Infinitely divisible distribution; Probability inequality;
PMID:	38261930	Category:	Date Added:	2024-01-23
Dept Affiliation:	MATHSTATS 1 Business School, Chengdu University, Chengdu, 610106 China. 2 College of Mathematics, Sichuan University, Chengdu, 610065 China. 3 Department of Mathematics and Statistics, Concordia University, Montreal, H3G 1M8 Canada.

Description:

Let a, ß be positive real numbers and let $X_{α, β}$ be a Gamma random variable with shape parameter a and scale parameter ß. We study infimum values of the function $(α, β) \mapsto P {X_{α, β} \leq κ E [X_{α, β}]}$ for any fixed $κ > 0$ and the function $(α, β) \mapsto P {| X_{α, β} - E [X_{α, β}] | \leq \sqrt{Var (X_{α, β})}}$ . We show that ${inf}_{α, β} P {X_{α, β} \leq E [X_{α, β}]} = \frac{1}{2}$ and ${inf}_{α, β} P {| X_{α, β} - E [X_{α, β}] | \leq \sqrt{Var (X_{α, β})}} = P {| Z | \leq 1} \approx 0.6827$ , where Z is a standard normal random variable.

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