Let α , β be positive real numbers and let X α , β be a Gamma random variable with shape parameter α and scale parameter β . We study infimum values of the function ( α , β ) ↦ P { X α , β ≤ κ E [ X α , β ] } for any fixed κ > 0 and the function ( α , β ) ↦ P { | X α , β - E [ X α , β ] | ≤ Var ( X α , β ) } . We show that inf α , β P { X α , β ≤ E [ X α , β ] } = 1 2 and inf α , β P { | X α , β - E [ X α , β ] | ≤ Var ( X α , β ) } = P { | Z | ≤ 1 } ≈ 0.6827 , where Z is a standard normal random variable.
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