The infimum values of two probability functions for the Gamma distribution.

Let α, ß be positive real numbers and let Xα,ß be a Gamma random variable with shape parameter α and scale parameter ß. We study infimum values of the function (α,ß)↦P{Xα,ß≤κE[Xα,ß]} for any fixed κ>0 and the function (α,ß)↦P{|Xα,ß-E[Xα,ß]|≤Var(Xα,ß)}. We show that infα,ßP{Xα,ß≤E[Xα,ß]}=12 and infα,ßP{|Xα,ß-E[Xα,ß]|≤Var(Xα,ß)}=P{|Z|≤1}≈0.6827, where Z is a standard normal random variable.