  One of the most popular models for quantitatively understanding the emergence
of drug resistance both in bacterial colonies and in malignant tumors was
introduced long ago by Luria and Delbr\"uck. Here, individual resistant mutants
emerge randomly during the birth events of an exponentially growing sensitive
population. A most interesting limit of this process occurs when the population
size $N$ is large and mutation rates are low, but not necessarily small
compared to $1/N$. Here we provide a scaling solution valid in this limit,
making contact with the theory of Levy $\alpha$-stable distributions, in
particular one discussed long ago by Landau. One consequence of this
association is that moments of the distribution are highly misleading as far as
characterizing typical behavior. A key insight that enables our solution is
that working in the fixed population size ensemble is not the same as working
in a fixed time ensemble. Some of our results have been presented previously in
shortened form.
