A drunkard's walk_Solution

A drunkard's walk
=====================

A drunkard begins walking aimlessly, starting at a lamp post. At each time step,
the drunkard forgets where he or she is, and takes one step at random, either
north, east, south, or west, with probability 25%. How far will the drunkard be
from the lamp post after `N` steps?

* Write a program `RandomWalker.java` that takes an integer command-line
argument `N` and simulates the motion of a random walker for `N` steps. After
each step, print the location of the random walker, treating the lamp post as
the origin `(0, 0)`. Also, print the square of the final distance from the
origin.

```
% java RandomWalker 10
(0, -1)
(0, 0)
...
6 more steps
...
(-2, 1)
(-3, 1)
squared distance = 10
```

```
% java RandomWalker 20
(0, 1)
(-1, 1)
...
16 more steps
...
(-3, -2)
(-3, -3)
squared distance = 18
```

* Write a program `RandomWalkers.java` that takes two integer command-line
arguments `N` and `T`. In each of `T` independent experiments, simulate a random
walk of `N` steps and compute the squared distance. Output the mean squared
distance (the average of the `T` squared distances).

```
% java RandomWalkers 100 100000
mean squared distance = 100.15086
```

```
% java RandomWalkers 400 100000
mean squared distance = 401.22024
```

```
% java RandomWalkers 100 100000
mean squared distance = 99.95274
```

```
% java RandomWalkers 800 100000
mean squared distance = 797.5106
```

```
% java RandomWalkers 200 100000
mean squared distance = 199.57664
```

```
% java RandomWalkers 1600 100000
mean squared distance = 1600.13064
```

*Remark: this process is a discrete version of a natural phenomenon known as*
*Brownian motion. It serves as a scientific model for an astonishing range of*
*physical processes from the dispersion of ink flowing in water, to the*
*formation of polymer chains in chemistry, to cascades of neurons firing in the*
*brain.*

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