The problem of random intervals on a line

Abstract

Suppose that events occur at random points on a line from t = −∞ to +∞, the probability of an event occurring between t and t + dt being λdt. If we select any interval of the line, say the interval [0, y], there will be a finite probability that it contains 0, 1, 2,…,r,…events; in fact, it is not difficult to show that these probabilities form a Poisson distribution, the probability that the interval contains r events being (see e.g. (1)). Consider the case when each event consists of an interval of length α (an event being characterized by its first point). What is the probability that the covered portion of the interval [0, y] lies between x and x + dx?