Other points concerning the solution

Some nice points emerge from the algebra. The Equations can be written more simply as $\sum$_i=1ⁿt_iA_ji=0∀j

Also one can replace $d$_if_i by $1+(A$_ji-a_ji)/p_jA_ji , from Equation , and the equations then reduce to $\sum$_iA_ji= ∑_ia_ji∀j

These are telling us that the estimates of the $A$_ji for some source will change the shape of the distribution from that of the MC data $a$_ji , but will not change the overall total number.

Equation can be multiplied by $A$_ji and summed over j to give

$\sum$_jd_i- p_jA_ji+ a_ji- A_ji=0

summing over i, and using Equation , gives

$\sum$_id_i= ∑_i∑_jp_ja_ji

$N$_D= ∑_jp_jN_j

which nicely returns the normalisation, and makes clear the significance of the $p$_j . It is interesting that such an automatic normalisation does not occur in the $\chi 2$ minimisation technique of Equation . If the different $p$_j are allowed to float independently they return a set of values for which the fitted number of events is generally less than the actual total number, as downward fluctuations have a smaller assigned error and are given higher weight.