Erlang probability distribution equivalent to GammaDistribution with a rate parameter
ErlangDistribution[k, λ]
Mean[ErlangDistribution[k, λ]]
→ k/λPDF[ErlangDistribution[k, λ], x]
→ Piecewise[{{x^(-1 + k)/(E^(x*λ)*λ^(-1)^k*Gamma[k]), x > 0}}, 0]CDF[ErlangDistribution[k, λ], x]
→ Piecewise[{{GammaRegularized[k, 0, x*λ], x > 0}}, 0]