  We show that galaxy ellipticity estimation for weak gravitational lensing
with unweighted image moments reduces to the problem of measuring a combination
of the means of three independent normal random variables. Under very general
assumptions, the intrinsic image moments of sources can be recovered from
observations including effects such as the point-spread function and
pixellation. Gaussian pixel noise turns these into three jointly normal random
variables, the means of which are algebraically related to the ellipticity. We
show that the random variables are approximately independent with known
variances, and provide an algorithm for making them exactly independent. Once
the framework is developed, we derive general properties of the ellipticity
estimation problem, such as the signal-to-noise ratio, a generic form of an
ellipticity estimator, and Cram\'er-Rao lower bounds for an unbiased estimator.
We then derive the unbiased ellipticity estimator using unweighted image
moments. We find that this unbiased estimator has a poorly behaved distribution
and does not converge in practical applications, but demonstrates how to derive
and understand the behaviour of new moment-based ellipticity estimators.
