  A study of multiplicity and pseudorapidity distributions of inclusive photons
measured in pp and p$-$Pb collisions at a center-of-mass energy per
nucleon$-$nucleon collision of $\sqrt{s_{\rm NN}} = 5.02$ TeV using the ALICE
detector in the forward pseudorapidity region $2.3 < \eta_{\rm lab} < 3.9$ is
presented. Measurements in p$-$Pb collisions are reported for two beam
configurations in which the directions of the proton and lead ion beam were
reversed. The pseudorapidity distributions in p$-$Pb collisions are obtained
for seven centrality classes which are defined based on different event
activity estimators, i.e., the charged-particle multiplicity measured at
midrapidity as well as the energy deposited in a calorimeter at beam rapidity.
The inclusive photon multiplicity distributions for both pp and p$-$Pb
collisions are described by double negative binomial distributions. The
pseudorapidity distributions of inclusive photons are compared to those of
charged particles at midrapidity in \pp collisions and for different centrality
classes in p$-$Pb collisions. The results are compared to predictions from
various Monte Carlo event generators. None of the generators considered in this
paper reproduces the inclusive photon multiplicity distributions in the
reported multiplicity range. The pseudorapidity distributions are, however,
better described by the same generators.
