The full-information best choice problem with a random number of observations

The full-information best choice problem with a random number of observations is considered. N i.i.d. random variables with a known continuous distribution are observed sequentially with the object of selecting the largest. Neither recall nor uncertainty of selection is allowed and one choice must be made. In this paper the number N of observations is random with a known distribution. The structure of the stopping set is investigated. A class of distributions of N (which contains in particular the uniform, negative-binomial and Poisson distributions) is determined, for which the so-called “monotone case” occurs. The theoretical solution for the monotone case is considered. In the case where N is geometric the optimal solution is presented and the probability of winning worked out. Finally, the case where N is uniform is examined. A simple asymptotically optimal stopping rule is found and the asymptotic probability of winning is obtained.