Simulation of Erlang and negative binomial distributions using the generalized Lambert W function

We present a simulation method for generating random variables from Erlang and negative binomial distributions using the generalized Lambert $W$ function. The generalized Lambert $W$ function is utilized to solve the quantile functions of these distributions, allowing for efficient and accurate generation of random variables. The simulation procedure is based on Halley’s method and is demonstrated through the generation of 100,000 random variables for each distribution. The results show close agreement with the theoretical mean and variance values, indicating the effectiveness of the proposed method. This approach offers a valuable tool for generating random variables from Erlang and negative binomial distributions in various applications.