  In this paper we propose estimators of the distribution of events of
different kinds in a multimode Poisson process. We give the explicit solution
for the maximum likelihood estimator, and derive its strong consistency and
asymptotic normality. We also provide an order restricted estimator and derive
its consistency and asymptotic distribution. We discuss the application of the
estimator to the detection of neutrons in a novel detector being developed at
the European Spallation Source in Lund, Sweden. The inference problem gives
rise to Sylvester-Ramanujan system of equations.
