The Problems of Random Flight and Conduction of Heat

Extract

In a paper on random flight Lord Rayleigh proved the following result: A number is formed by adding together n numbers each of which is equally likely to have any value from − a to + a. Then, if f (n, s) ds is the probability that the number so formed lies between s and s + ds, and if n is sufficiently great,

This result may be stated as follows: A point moves discontinuously in a straight line. For a time τ it has a constant velocity. During the next time-interval τ it again has a constant velocity, and so on. Then if each of these velocities is equally likely to have any value from − v to + v, the probability that in the time nτ, the point moves a distance lying between s and s + ds is f (n, s) ds, with vτ written for a.