Gamma function and normalisation

Γ(r)=∫ from 0 to ∞ t raised to the power of r-1 ^r-1exp(-t)dt for r>0; Γ(r+1)=rΓ(r) and Γ(m)=(m-1)! for positive integers. With shape r>0 and rate λ>0, the gamma density is λ^r x raised to the power of r-1 ^r-1exp(-λx)/Γ(r), x>0. Substitution t=λx makes its total integral equal to 1. The supplement uses the rate convention; a scale parameter would be 1/λ.