 Rock Street, San Francisco

1. What are Random
Walks?

1.1
A Random Walk

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A random walk is a
stochastic process by which randomly-moving objects wander away from where they
started. An example of a random walk is the random walk on the integer number
line, which starts at 0 and at each step moves +1 or ?1 with equal probability 17.

1.2 Variables that govern a random walk 13, 14, 15

·
Uniformity:
If a set of random numbers between 0 and 6 is generated, the first number in
the sequence to appear is equally likely to be 0, 1, 2, . . . ,6. Additionally,
any number i in the sequence has the
equal possibility to be 0, 1, 2, . . . ,6. The average of the numbers should be
3.5.

·
Independence:
The values must be independent of each other. If it becomes possible to predict
a value in the sequence, based on the previous value, then the process is not
random. Hence, the probability of observing each value is not dependent upon
previous values.

·
Summation:
Sum of two consecutive numbers has the equal possibility to be even or odd.

·
Duplication:
When a thousand numbers a generated randomly, it is possible that some will be
duplicated. If 1,000 numbers are randomly generated, some will be duplicated.

Throwing a fair
dice with 1, 2, 3, 4, 5 or a 6 on its top face resembles a random walk. Each
throw has the probability of 1/6 of generating any of the six numbers. Again,
the throw is independent of each other and the result of one throw does not
have any impact of the next throw
regardless of how many elements have already been produced. Therefore, the
unbiased “fair” dice is thus the perfect example of generating random number, since the 1 to 6 will be
distributed randomly.

1.3 Effect of Number of Walks on Simulation 18

The effect of the
number of walk on the simulation result can be described by coin flipping game.
If it is head, then move +1 and if tail
move -1. So, after one flip, there is ½ probability to be on +1 & -1. After
two flips, there is 1/4 probability to be on +2 & -2 and ½ probability to
be on 0. Again, after three flips, there is 1/8 probability to be on +3 &
-3, 3/8 probability to be on +1 & -1and ½ probability to be on 0. So, it is
observed that it mostly hovers around the middle i.e. 0. As the coin is flipped
longer the more it spreads out. So, the probability of random walk of being in
a specific position starts to create a bell curve.

Fig 1: Number Line

1.4
Algorithm for the Dice Problem

1.
Initialize
variable: for storing the count of 5; assign number of walks (toss of dice)

2.
Repeat
simulation of random walk in the form of dice tossing 1000 times

3.
Initialize
a random number generator with numbers from 1 to six and store them

4.
If
the face of the dice after tossing is equal to 5, collect the count.

5.
Find
the probability of by dividing the number of counts by the tossing number

6.
Plot
the probability distribution of getting 5 i.e. Probability vs. Toss

7.
Compute
and print the estimated probability

1.5
Results of Fair Dice Problem

Probability for 1000 random walks = 0.171

Theoretical probability = 0.167

1.6 Discussion

The plot of the
probability as a function of a number of
‘walks walked’ is added. The plot shows that the probability is high when dice
throwing is started. This is because of getting 5s near the beginning. To get the probability, the number of 5 got is divided by the number
thrown, So, the theoretical probability is 1/6 for a fair dice. But for random
walk initially, the number of times
thrown is little compared to the number of times 5 shows. So, the probability
distribution curve shows an elevation. As the number of random walks increases it resemble theoretical result which
is evident from the above plot.

2.
Few applications
of random walks

2.1 Simulation

Simulation
is defined as the replication of the real-world operation over time 1. To
simulate something a model needs to be developed which constitutes the basic characteristics
and functions of the operation. The model represents the system itself, whereas
the simulation represents the operation of the system over time 2. Models can
be both deterministic and stochastic.

2.2 Deterministic Model

A
deterministic model is one that has no stochastic element. A process is
deterministic if its future is completely determined by its present and past.
The models are mostly described by differential equations in this case.
Different numerical methods are implemented to solve the equations: finite
difference, path simulation 3.

2.3 Stochastic Model

A
stochastic model represents a situation where uncertainty is present. In other
words, it’s a model for a process that has randomness. In this model unique
input leads to a different output for
each model run. Because of the randomness of the components, each simulation gives only one result 5.

2.4 When Random Walk Simulation is
Preferred?

A
random walk is a stochastic or random process, that defines a path that
consists of a series of random steps on a mathematical space 4. A random walk
process can’t be solved analytically. A random walk model can be simulated by
generating a random number to execute trial. Such a simulation is called ‘Monte Carlo Method’.  Deterministic simulation is one kind of random
walk simulation.

In
the deterministic model, there is no
random element and simulation can be done just once. A random walk model can
handle the uncertainty in the inputs built into it. Therefore, random walk
model should be used if there are uncertainties in the inputs and the process
is not fully known 5.

With
just a few cases, the deterministic analysis
makes it difficult to see which variables impact the outcome the most. In Monte
Carlo simulation, it’s easy to see which inputs had the biggest effect on
bottom-line results.

Again,
in deterministic models, modeling different combinations of values for
different inputs to see the effects of truly different scenarios is very
difficult. Using Monte Carlo simulation, analysts can observe clearly which
inputs had which values together when certain outcomes occurred. This is
invaluable for pursuing further analysis 6.

Simulation 6, 7

·
The model is can be flexibly used
for probability distributions

·
Models can be developed for
correlations and other relations

·
Again, independent relationships
can be established between input variables.

·
The behavior of and changes to the
model can be investigated with great ease and speed.

·
The level of mathematics required
is quite basic

Simulation 6 8

·
Typically, a computer is required

·
Sometimes, calculations can be time-consuming than analytical models

·
Solutions are approximate but not
exact and depend on the number of repeated runs used to produce the output
statistics.

·
There are no limits on the error of
the computed result. 