Skew probability curves with negative powers of time and related to random walks in series

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Abstract
Summary The well known probability distribution of first arrival times of a particle undergoing random walk or Brownian movement in one dimension is extended to allow for steps in series each in a different medium. Previously this led to considering a certain distribution defined by its cumulants, which form a simple series generalising that for the known distribution. This is illustrated by the particular case of two first passages in series. Approximations to the probability (density) curves are found, each of which consists of a sharp peak followed by a long tail where the ordinates are very nearly proportional to t‐W, W ≥ 3/2. A generalisation can yield smaller W down to ca. 0.1. It is concluded that this explains why negative powers of time are found in so many physiological clearance curves of all kinds. Numerical tables are based on distributions with very smal step times, and give ones built up from the sum of a varying number of steps. The parameters 01 the triangle formed by the inflection tangents are given in order to describe the peaks.