The Problem of the Random Walk

Can any of your readers refer me to a work wherein I should find a solution of the following problem, or failing the knowledge of any existing solution provide me with an original one? I should be extremely grateful for aid in the matter.

A man starts from a point O and walks l yards in a straight line; he then turns through any angle whatever and walks another l yards in a second straight line. He repeats this process n times. I require the probability that after these n stretches he is at a distance between r and r + dr from his starting point, O.

The problem is one of considerable interest, but I have only succeeded in obtaining an integrated solution for two stretches. I think, however, that a solution ought to be found, if only in the form of a series in powers of 1/n, when n is large.

KARL PEARSON.

The Gables, East Ilsley, Berks.

The Problem of the Random Walk

This problem, proposed by Prof. Karl Pearson in the current number of Nature, is the same as that of the composition of n iso-periodic vibrations of unit amplitude and of phases distributed at random, considered in Phil. Mag., x., p. 73, 1880; xlvii., p. 246, 1899; ("Scientific Papers," i., p. 491, iv., p. 370). If n be very great, the probability sought is

Probably methods similar to those employed in the papers referred to would avail for the development of an approximate expression applicable when n is only moderately great.

RAYLEIGH.

Terling Place, July 29.

The Problem of the Random Walk

I have to thank several correspondents for assistance in this matter. Mr. G. J. Bennett finds that my case of n=2 can really be solved by elliptic integrals, and, of course, Lord Rayleigh's solution for n very large is most valuable, and may very probably suffice for the purposes I have immediately in view. I ought to have known it, but my reading of late years has drifted into other channels, and one does not expect to find the first stage in a biometric problem provided in a memoir on sound. From the purely mathematical standpoint, it would still be very interesting to have a solution for n comparatively small. The sections through the axis of Lord Rayleigh's frequency surface for n large are simply the "cocked hat" or normal curve of errors type, for n=2 or 3 they do not resemble this form at all. For n=2, for example, the sections are of the form of a double U, thus UU, the whole being symmmetrical about the centre vertical corresponding to r=o, but each U istelf being asymmetrical. The system has three vertical asymptotes. It would be interesting to see how the multicity of types of n small passes over into the normal curve of errors when n is made large.

The lesson of Lord Rayleigh's solution is that in open country the most probable place to find a drunken man who is at all capable of keeping on his feet is somewhere near his starting point!

KARL PEARSON

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