ON THE RANGE OF PARTIAL SUMS OF A FINITE NUMBER OF INDEPENDENT NORMAL VARIATES

The present investigation was suggested by the problem of planning the storage capacity of a reservoir, where one would like to know the distribution of the water level over a given number n of years. The level after r years may be regarded as the sum of r annual increments, and we may approximate this real problem by the ideal one where the annual increments are independent variates with a common distribution. Applications to other storage problems are obvious. In the case of very large n the asymptotic range of the consequent random walk has been discussed by Feller (1951). In the present paper we are concerned with an exact formulation for the case of finite n, taking the increments to be standard normal variates, and for this case we obtain a simple expression for the mean range. The paper falls into two parts. In the first (??2-5), tlle problem is reduced to that of evaluating the integral