Statistics Of Microlensing Optical Depth

We show that the statistical error, $\sigma_{\tau}$, in estimating the optical depth, $\tau$, due to microlensing is substantially higher than the naive Poisson estimate: $\sigma_{\tau} = \sqrt{\eta / N} \tau$, where $N$ is the number of detections and $1.65 \leq \eta \lsim 3$. We construct a ``microlensing map'' using an observationally well-constrained barred bulge model and compare this with a map based on a spherical bulge model. The result indicates a significant differences in the distribution of optical depths. If the optical depths are determined over a wide range of positions around the Galactic center, one could make a rudimentary determination of the geometry of the mass distribution of the Galactic bulge by observing $\sim 200$ events. About 800 events are required to determine the relative contributions to the event rate by bulge and disk Machos.