different direction at random and walks a distance Yor He then 
walks a distance Y3 in another direction picked at random, and 
so on. The problem is to determine the probability that he will 
be within a distance R from the pub if the total distance he 
has walked is given by Y = ¥, + Yo + seeeet Vos and if his choice 
of directions has been completely random. 
The solution to the random walk problem as Y becomes larger 
and the Vy shrink smaller and smaller is the normal probability, 
or Gaussian, distribution. From the description of the problem, 
it would appear that the drunkard would not end up too far away 
from the pub. A whole statistical class of wayfarers would show 
most of them concentrated near the origin and a few scattered 
at greater distances away. The extension of the random walk 
problem into three dimensions is the problem of Brownian motion 
and similar results are obtained. 
The connection of the random walk problem with equation 
(7.232) is that in the vector notation shown the partial sums are 
all basically random walks. The projection of the sum of the 
vectors on the real axis is also a Gaussian distribution in the 
limit as r approaches infinity andA»/# approaches zero. Equa- 
tion (7.33) is a Consequence of this result. It states that 
the probability, at a time, ty> chosen at random, that the ampli- 
tude of the sea surface will be less than the value K is given 
by the normal probability distribution. K is the departure from 
the mean, assumed to be zero, of the record. Equation (7.34) 
is another way to express this condition. It gives the probabi- 
lity that a point chosen at random in the record will lie between 
- 139 - 
