  With $X^*$ denoting a random variable with the $X$-size bias distribution,
what are all distributions for $X$ such that it is possible to have $X^*=X+Y$,
$Y\geq 0$, with $X$ and $Y$ {\em independent}? We give the answer, due to
Steutel \cite{steutel}, and also discuss the relations of size biasing to the
waiting time paradox, renewal theory, sampling, tightness and uniform
integrability, compound Poisson distributions, infinite divisibility, and the
lognormal distributions.
