rank 437

2

2

second problem, however, in that in a truly random

sequence, numbers do indeed appear more than once,

which is avoided here.) Many computers today use this

method with the choice m= 231.

Computer-generated pseudo-random numbers are

used in

STATISTICS

to help facilitate unbiased sampling

of a

POPULATION

.

See also

SAMPLE

.

random variable See

DISTRIBUTION

.

random walk In the theory of statistical mechanics,

a random walk is the path traced out by a particle per-

forming a sequence of unit steps in which the direction

of each step is chosen randomly. For example, gas

molecules randomly jolting back and forth in space are

effectively performing three-dimensional random

walks. (Their resultant motion is called Brownian

motion, which can be used to explain diffusion.)

A child standing on a sidewalk can perform a one-

dimensional random walk by taking steps backward

and forward according to the results of tossing a coin

(say, “heads” means step forward one pace, “tails” step

backward one pace). Another child standing in an infi-

nite grid of squares performs a two-dimensional ran-

dom walk by stepping one place north, south, east, or

west according to the rolls of a four-sided die (or of an

ordinary die if rolls of 5 and 6 are ignored).

Gamblers betting $1 at a time at a casino are effec-

tively performing random walks—the contents of their

wallets increase and decrease in a random fashion, one

unit at a time.

We can ask, what are the chances that a gambler

will eventually lose all of his or her money? (Or equiva-

lently, what is the probability that a particle performing

a one-dimensional random walk will eventually move

one place behind “start”?)

This probability can be computed as follows. Sup-

pose the gambler has Ndollars in hand and, at each

round of play, has a 50 percent chance of going up $1

and a 50 percent chance of losing $1. Let pbe the

probability that the gambler will eventually only have

N– 1 dollars in hand.

There are two ways for a gambler to lose this dol-

lar, either right away, or to win $1 and then eventually

lose $2 later on. This shows:

To lose $2, a gambler must first lose $1 and then

another. Thus:

We thus have the equation: , that is,

(p–1)

2= 0, showing that p= 1. That is, with absolute

certainty, a gambler will eventually go down $1, and

then another, and then another, all the way down to

ruin. This phenomenon is called

GAMBLER

’

S RUIN

.

The above calculation shows that all one-dimen-

sional random walks (with equal chances of stepping

backward or forward) eventually return to their start-

ing positions. Mathematicians have shown that two-

dimensional random walks (again with equal chances

of stepping in any one of four directions) also eventu-

ally return to their starting positions with absolute cer-

tainty. Surprisingly, there is only a 34 percent chance

that a three-dimensional random walk will have the

particle return to its starting position.

See also

HARMONIC FUNCTION

.

rank In

STATISTICS

, an arrangement of a set of objects

or

DATA

values in an order dictated by the magnitude

or importance of some characteristic is called a ranking

of the objects. For example, the arrangement of 20 men

in order of height is a ranking, as is the arrangement of

10 numbers from lowest to highest. The position of an

object in a list given by a ranking is called the rank of

the object. A ranking could be purely subjective, such