  We consider a linear ill-posed equation in the Hilbert space setting.
Multiple independent unbiased measurements of the right hand side are
available. A natural approach is to take the average of the measurements as an
approximation of the right hand side and to estimate the data error as the
inverse of the square root of the number of measurements. We calculate the
optimal convergence rate (as the number of measurements tends to infinity)
under classical source conditions and introduce a modified discrepancy
principle, which asymptotically attains this rate.
