Gaussian profile estimation in one dimension

We present several new results on the classic problem of estimating Gaussian profile parameters from a set of noisy data, showing that an exact solution of the maximum likelihood equations exists for additive Gaussian-distributed noise. Using the exact solution makes it possible to obtain analytic formulas for the variances of the estimated parameters. Finally, we show that the classic formulation of the problem is actually biased, but that the bias can be eliminated by a straightforward algorithm.