Methodology.

The correct way to view the problem is as follows. For each source, in each bin, there is some (unknown) expected number of events $A$_ji . The prediction for the number of data events in a bin is not Equation but $f$_i= ∑_j=1^mp_jA_ji

From each $A$_ji the corresponding $a$_ji is generated by a distribution which is in fact binomial, but can be taken as Poisson if $A$_ji<<N_j (which is indeed the case, as our problem is just that a large number of total events gives a small number in each bin.)

The total likelihood which is to be maximised is now the combined probability of the observed $d$_i and the observed $a$_ji

and we want to maximise $lnL=\; \sum$_i=1ⁿd_ilnf_i- f_i+ ∑_i=1ⁿ∑_j=1^ma_jilnA_ji- A_ji

The estimates for the $p$_j (which we want to know) and the $A$_ji (in which we're not really interested) are found by maximising this likelihood. This is the correct methodology to incorporate the MC statistics: unfortunately it consists of a maximisation problem in $m\; x(n+1)$ unknowns. However, the problem can be made much more amenable.