  The problem of fitting an event distribution when the total expected number
of events is not fixed, keeps appearing in experimental studies. In a
chi-square fit, if overall normalization is one of the parameters parameters to
be fit, the fitted curve may be seriously low with respect to the data points,
sometimes below all of them. This problem and the solution for it are well
known within the statistics community, but, apparently, not well known among
some of the physics community. The purpose of this note is didactic, to explain
the cause of the problem and the easy and elegant solution. The solution is to
use maximum likelihood instead of chi-square. The essential difference between
the two approaches is that maximum likelihood uses the normalization of each
term in the chi-square assuming it is a normal distribution, 1/sqrt(2 pi
sigma-square). In addition, the normalization is applied to the theoretical
expectation not to the data. In the present note we illustrate what goes wrong
and how maximum likelihood fixes the problem in a very simple toy example which
illustrates the problem clearly and is the appropriate physics model for event
histograms. We then note how a simple modification to the chi-square method
gives a result identical to the maximum likelihood method.
