English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Motivated by Chv\'{a}tal's conjecture and Tomaszewaki's conjecture, weinvestigate the extreme value problem of two probability functions for theGamma distribution. Let $\alpha,\beta$ be arbitrary positive real numbers and$X_{\alpha,\beta}$ be a Gamma random variable with shape parameter $\alpha$ andscale parameter $\beta$. We study the extreme values of functions$P\{X_{\alpha,\beta}\le E[X_{\alpha,\beta}]\}$ and$P\{|X_{\alpha,\beta}-E[X_{\alpha,\beta}]|\le \sqrt{{\rmVar}(X_{\alpha,\beta})}\}$. Among other things, we show that $\inf_{\alpha,\beta}P\{X_{\alpha,\beta}\le E[X_{\alpha,\beta}]\}=\frac{1}{2}$and $\inf_{\alpha,\beta}P\{|X_{\alpha,\beta}-E[X_{\alpha,\beta}]|\le \sqrt{{\rmVar}(X_{\alpha,\beta})}\}=P\{|Z|\le 1\}\approx 0.6826$, where $Z$ is a standardnormal random variable.

The extreme values of two probability functions for the Gamma distribution