Characterization of Distributions with the Length-bias Scaling Property

For q ∈ (0,1) fixed, we characterize the density functions f of absolutely continuous random variables X > 0 with finite expectation whose respective distribution functions satisfy the so-called (LBS) length-bias scaling property X L = qb X , where b X is a random variable having the distribution function b F (x) = (EX )−1 ∫ x 0 y f y d y. For an absolutely continuous random variable X > 0 with probability density function (pdf) f and finite expectation EX , we denote by b X an absolutely continuous random variable having the probability density function (EX )−1 x f (x) . In this case, b X is called the sizeor length-biased version of X and L (b X ) is the corresponding length-biased distribution. It is well known that b X is the stationary total lifetime in a renewal process with generic lifetime X (see [2, Chapter 5]). The length-biased distributions have been applied in various fields, such as biometry, ecology, environmental sciences, reliability and survival analysis. A review of these distributions and their applications are included in [5, Section 3], [6, 8, 12, 13]. In [9], Pakes and Khattree ask whether it is possible to randomly rescale the total lifetime to recover the lifetime law. More specifically, let V ≥ 0 be a random variable independent of X with a fixed law satisfying P (V > 0)> 0. For which laws L (X ) does the following “equality in law” X L = V b X , hold? For instance, when V has the uniform law on [0,1] the last equality holds if and only if L (X ) is an exponential law (see [9]). In this note we consider the case where V is a constant function: The law of X has the so-called length-bias scaling property (abbreviated to LBS-property) if