Characterization of distributions with the length-bias scaling property

Marcos Lopez-Garcia

doi:10.1214/ecp.v14-1458

Abstract

This paper characterizes the density functions of absolutely continuous positive random variables with finite expectation whose respective distribution functions satisfy the so-called length-bias scaling property.

Highlights

For an absolutely continuous random variable X > 0 with probability density function f and finite expectation EX, we denote by X an absolutely continuous random variable having the probability density function (EX )−1 x f (x)
From Proposition 1 we obtain the characterization of the probability density functions with the LBS-property
By (3), it follows that the probability density functions having the LBS-property are solutions of an indeterminate moment problem

Summary

Introduction

For an absolutely continuous random variable X > 0 with probability density function (pdf) f and finite expectation EX , we denote by X an absolutely continuous random variable having the probability density function (EX )−1 x f (x). Let V ≥ 0 be a random variable independent of X with a fixed law satisfying P (V > 0) > 0. When X is an absolutely continuous random variable with probability density function f , we sometimes write X ∼ f .

Results

Conclusion