Think of the random walk as a game, where the player starts at the origin (i.e. all coordinates equal ) and at each move, he is required to make one step on an arbitrarily chosen axis. For example, in two dimensions, the player would step forwards, backwards, left, or right. The choice is to be made randomly, determined, for instance, by the toss of a fair coin. This is related to many other random events such as Brownian motion, and combination of error in measurements.
Expected Progress in One Dimension
Basic probability would tell you that the expected deviation from the origin in the long run ought to be zero. But one must have the feeling that as , the number of steps taken increases, he is more likely to have strayed farther away from the origin, . So, we might ask, what is the average of , where is the net distance (from the origin) traveled in steps.
After one step, the value of . The expected value of , for , can be obtained from , as
, and because we expect to reach each one of the values half the time, the average of is just going to be.
is also called root mean squared (rms) deviation and in similar situations to ours, we can expect the absolute value of the deviation of an observed value from the mean, in our case the position x=0, to be less than or equal to the rms.
Example Question 1
What would be the width of the range into which the experimental probability should fall so that the coin tossed can be considered a fair one?
Examples in one dimension:
, each time jumping to one of the adjacent vertices with equal probability. Let Sally start her daily workout in , and a mine be located in . Every second Sally must make her random jump (as described above). What is Sally's expected lifespan, in minutes, in this system?A frog, namely, Sally, is jumping about the vertices of a hexagon,
As a bonus, can you generalise for jumps?
This problem is a part of my froggy, soggy set.
Here, we apply the Random Walk in - dimensional space.
A drunkard walks out of a bar at midnight. He is in 3-dimensional space of an infinite size. The probability of his stepping 1 step forward, backward, left, right, up or down is equal. Calculate the probability (to two decimal places) of his (eventually) returning to the bar.
Details and Assumptions:
Assume the bar is a point particle at the origin.
We require the probability that he will, at any point in time, return to the bar.
Do not try this at home.