<p><it>Abstract—</it> Fault coverage and test length estimation in circuits under random test is the subject of this paper. Testing by a sequence of random input patterns is viewed as sequential sampling of faults from a given fault universe. Based on this model, the probability mass function ($<tmath>\mbi{pmf}</tmath>$) of fault coverage and expressions for all its moments are derived. This provides a means for computing estimates of fault coverage as well as determining the accuracy of the estimates. </p><p>Test length, viewed as waiting time on fault coverage, is analyzed next. We derive expressions for its $<tmath>\mbi{pmf}</tmath>$ and its probability generating function ($<tmath>\mbi{pgf}</tmath>$). This allows computation of all the higher order moments. In particular, expressions for mean and variance of test length for any specified fault coverage are derived. This is a considerable enhancement of the state of the art in techniques for predicting test length as a function of fault coverage. It is shown that any moment of test length requires knowledge of all the moments of fault coverage, and hence, its $<tmath>\mbi{pmf}</tmath>$. For this reason, expressions for approximating its expected value and variance, for user specified error bounds, are also given. A methodology based on these results is outlined. Experiments carried out on several circuits demonstrate that this technique is capable of providing excellent predictions of test length. Furthermore it is shown, as with fault coverage prediction, that estimates of variances can be used to bound average test length quite effectively.</p><p><it>Index Terms—</it>Fault coverage, test length, urn models, occupancy, waiting time.</p>