  We extend the theory of shot noise in coherent metals to shot noise in
strange metals without quasiparticle excitations. This requires a
generalization of the Boltzmann equation with a noise source to distribution
functions which depend independently on the excitation momentum and energy. We
apply this theory to a model of a strange metal with linear in temperature
($T$) resistivity, describing a Fermi surface with a spatially random Yukawa
coupling to a critical boson. We find a suppression of the Fano factor in the
strange metal, and describe the dependence of the shot noise on temperature and
applied voltage. At low temperatures, we obtain a Fano factor equal to $1/6$,
in contrast to the $1/3$ Fano factor in diffusive metals with quasiparticles.
Our results are in general agreement with recent observations by Chen et al.
(arXiv:2206.00673). We further compare the random Yukawa model to quasi-elastic
electron-phonon scattering that also generates $T$-linear resistivity, and
argue that shot noise observations offer a useful diagnostic to distinguish
between them.
