Laplace and Mellin transform for reconstructing the probability distribution by a limited amount of information

Lizhi Niu, Mario Di Paola, Antonina Pirrotta, Wei Xu

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

A method for reconstructing the Probability Density Function (PDF) of a random variable using the Laplace transform is first introduced for one-sided PDFs. This approach defines new complex quantities, referred as Shifted Characteristic Functions, which allow the PDF to be computed using a classical Fourier series expansion. The method is then extended to handle double-sided PDFs by redefining the double-sided Laplace transform. This new definition remains applicable even when the integral in the inverse Laplace transform is discretized along the imaginary axis. For comparison, a new definition of double-sided Complex Fractional Moments based on Mellin transform is also introduced, addressing the singularity at zero that arises during PDF reconstruction.

Original languageEnglish
Article number103700
JournalProbabilistic Engineering Mechanics
Volume78
DOIs
StatePublished - Oct 2024

Keywords

  • Complex fractional moment
  • FPK equation
  • Fourier transform
  • Laplace transform
  • Shift characteristic function

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