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where <Pi , <p2 , ■ ■ • , <pn are independent random angles, each distributed uniformly over the range — x to ir. z cannot exceed n. When n — 2 < z < n the probability density of z may be expressed as a power series in (n — z), as is shown in Sec. 2. There is a close relation between the distribution of z and the problem of the random walk in two dimensions, and the two are often treated together. Several equations connecting them are given in Sec. 3. In Sec. 4 the results of Sec. 2 are used to obtain the first few terms in a series for the distribution of the extreme values in the random walk problem. When n is large the central portion of the distribution for z approaches a normal law. In Sec. 5 an attempt is made to obtain an approximation to the distribution over the entire range of z by interpolating between the normal law result for small z and the results of Sec. 2 which hold for extreme values of z. The work is carried out first for the random walk distribution and then translated to the z distribution. This procedure is used because the random walk distribution seems to be better suited to our method of interpolation than does the z distribution. Figure 1 is associated with the interpolation between the results given by Pearson2 and Rayleigh3 for the random walk and those of Sec. 4. I wish to express my thanks for the many helpful suggestions concerning this paper which I have received from Mr. John Riordan and others. 2. Series for the probability density of z when z is near n. Let qn{z) denote the probability density of the random variable z defined by (1.1). Then, when n — 1 <

Distribution of the extreme values of the sum of 𝑛 sine waves phased at random