The classical Luria-Delbr\"uck model for fluctuation analysis is extended to
the case where cells can either divide or die at the end of their generation
time. This leads to a family of probability distributions generalizing the
Luria-Delbr\"uck family, and depending on three parameters: the expected number
of mutations, the relative fitness of normal cells compared to mutants, and the
death probability of mutants. The probabilistic treatment is similar to that of
the classical case; simulation and computing algorithms are provided. The
estimation problem is discussed: if the death probability is known, the two
other parameters can be reliably estimated. If the death probability is
unknown, the model can be identified only for large samples