Moment Monotonicity of Weibull, Gamma and Log-normal Distributions

This paper investigates the moment monotonicity property of Weibull, Gamma, and Log-normal distributions. We provide the first complete mathematical proofs for the monotonicity of the function $E(X^n)^{\frac{1}{n}}$ specific to these distributions. Through the derivations, we identify a key property: in many cases, one of the two parameters defining each distribution can be effectively canceled out. This finding opens up opportunities for improved parameter estimation of these random variables. Our results contribute to a deeper understanding of the behavior of these widely used distributions and offer valuable insights for applications in fields such as reliability engineering, econometrics, and machine learning.