  In this paper we derive an exact full expression for the 2D probability
distribution of the ellipticity of an object measured from data, only assuming
Gaussian noise in pixel values. This is a generalisation of the probability
distribution for the ratio of single random variables, that is well-known, to
the multivariate case. This expression is derived within the context of the
measurement of weak gravitational lensing from noisy galaxy images. We find
that the third flattening, or epsilon-ellipticity, has a biased maximum
likelihood but an unbiased mean; and that the third eccentricity, or normalised
polarisation chi, has both a biased maximum likelihood and a biased mean. The
very fact that the bias in the ellipticity is itself a function of the
ellipticity requires an accurate knowledge of the intrinsic ellipticity
distribution of the galaxies in order to properly calibrate shear measurements.
We use this expression to explore strategies for calibration of biases caused
by measurement processes in weak gravitational lensing. We find that upcoming
weak lensing surveys like KiDS or DES require calibration fields of order of
several square degrees and 1.2 magnitude deeper than the wide survey in order
to correct for the noise bias. Future surveys like Euclid will require
calibration fields of order 40 square degree and several magnitude deeper than
the wide survey. We also investigate the use of the Stokes parameters to
estimate the shear as an alternative to the ellipticity. We find that they can
provide unbiased shear estimates at the cost of a very large variance in the
measurement. The python code used to compute the distributions presented in the
paper and to perform the numerical calculations are available on request.
