Length Biasing and Laws Equivalent to the Log-Normal

Let X)0 denote a generic lifetime of a renewal process having unit mean lifetime, let Xˆ denote the stationary total lifetime and let qg 0, 1. be a fixed constant. We consider anew the scale invariance problem: For which laws does qXˆ have the same distribution as X? Our setting is more probabilistic than those presented hitherto, and we explore connections with the log-normal moment problem. In particular it is shown that all explicitly known laws which have log-normal moments solve our problem. The notion of remaining lifetime is generalized and its scaling invariance is investigated using the notion of total lifetime. Two moment.equivalent laws of Askey are shown to have a simple representation in terms of laws equivalent to the log-normal. The representation involves a q-gamma law, which we explore in its own right. An affine extension of our basic scale invariance relation, arising in the theory of orthogonal polynomials, is shown to be equivalent to the latter.