2 1. Random Walk and Discrete Heat Equation
Sn
1 3 4 6 5 2 7 8
0
-1
-2
3
2
1
-3
n
Figure 1. One-dimensional random walk with x = 0
What is the probability that at a particular time the walker
is at the origin?
More generally, what is the probability distribution for the
position of the walker?
Does the random walker keep returning to the origin or does
the walker eventually leave forever?
Probabilists use the notation E for expectation (also called ex-
pected value, mean, average value) defined for discrete random vari-
ables by
E[X] =
z
z P{X = z}.
The random walk satisfies E[Sn] = 0 since steps of +1 and −1 are
equally likely. To compute the average distance, one might try to
Previous Page Next Page