To understand the concept of Monte –Carlo Method and its various applications and it rely on repeated and random sampling to obtain numerical result. Developing the computational algorithms to solve the problem related to random sampling. Objective also contains simulation of specific problem in Matlab Software.

1. PROJECT BASED LEARNING : PROBABILITY AND RANDOM
PROCESSES
Study on Monte-Carlo Method
using MATLAB Code
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2. OBJECTIVE OF THE PROJECT
• To understand the concept of Monte –Carlo Method and its various applications
and it rely on repeated and random sampling to obtain numerical result.
• Developing the computational algorithms to solve the problem related to random
sampling.
• Objective also contains simulation of specific problem in Matlab Software.
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3. TERMINOLOGIES
1. Computational Algorithm.
2. Random Sampling.
3. Mathematical Optimization.
4. Probability Distribution.
Computational algorithm
An exactly defined specification of the operations to be carried out on data, by means
of which it is possible, using a discrete-operation digital computer, to convert a certain
amount of data (input data) into a certain amount of other data (output data) by
performing a finite number of operations. A computational algorithm is realized in the
form of a computational process, i.e. as a finite sequence of states of a real computer,
discretely distributed in time, the real computer unlike an abstract computer having a
restricted rate of performance of the operations, a restricted number of digit places to
form a number and a restricted storage capacity.
Random sampling
One of the best ways to achieve unbiased results in a study is through random
sampling. Random sampling includes choosing subjects from a population through
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4. unpredictable means. In its simplest form, subjects all have an equal chance of being
selected out of the population being researched.
Mathematical Optimization
In mathematics, computer science, or management science, mathematical
optimization (alternatively, optimization or mathematical programming) is the selection
of a best element (with regard to some criteria) from some set of available alternatives.
Probability Distribution
In probability and statistics, a probability distribution assigns a probability to each of the
possible outcomes of a random experiment, survey, or procedure of statistical
inference. Examples are found in experiments whose sample space is non-numerical,
where the distribution would be a categorical distribution; experiments whose sample
space is encoded by discrete random variables, where the distribution is a probability
mass function; and experiments with sample spaces encoded by continuous random
variables, where the distribution is a probability density function. More complex
experiments, such as those involving stochastic processes defined in continuous-time,
may demand the use of more general probability measures.

INTRODUCTION TO THE PROJECT
This project is a study to understand the concept of Monte-Carlo methods which are the
broad class of computational algorithm that rely on repeated random sampling to obtain
numerical results i.e. by running simulations many times over in order to calculate those
same probabilities heuristically just like actually playing and recording your results in a
real casino situation.
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5. They are often used in physical and mathematical problems and are most suited to be
applied when it is impossible to obtain a closed-form expression or infeasible to apply
a deterministic algorithm.
Monte Carlo methods are mainly used in three distinct problems:
1. Optimization.
2. Generation of samples from a probability distribution.
3. Numerical Integration.
Monte Carlo methods are especially useful for simulating systems with
many coupled degrees of freedom, such as:
• Fluids.
• Disordered materials.
• Strongly coupled solids.
• Cellular structures (cellular Potts model).
• Significant uncertainty in inputs, such as the calculation of risk in business.
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6. Note - We have mainly studied Monte-Carlo method in category of generation of
samples from a probability distribution.
Steps to generate a 'Monte-Carlo'
Step1 : Define a domain of possible inputs.
Step2 : Generate inputs randomly from a probability distribution over the
domain.
Step3 : Perform a deterministic computation on the inputs.
Step4 : Aggregate the results.
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7. Fig.1: Monte-Carlo method simulation step wise flow chart.
Calculation of Pi Using Monte-Carlo Method using
MATLAB
Problem Statement:
Consider a circle inscribed in a unit square. Given that the circle and the square have a
ratio of areas that is π/4, the value of π can be approximated using a Monte Carlo
method.
Solution:
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8. There are following steps involved for calculating the value of Pi using Monte-Caro
method as specified in the problem statement. High level language, Matlab is used to
tackle the problem by applying the algorithms of Monte-Carlo in following steps :
• Draw a square on the ground, and then inscribe a circle within it.
• Uniformly scatter some objects of uniform size (grains of rice or sand) over the
square.
• Count the number of objects inside the circle and the total number of objects.
• The ratio of the two counts is an estimate of the ratio of the two areas, which
is π/4. Multiply the result by 4 to estimate π.
If you are playing the dart, it is easy to imagine throwing darts randomly at Figure 2, and
it should be apparent that of the total number of darts that hit within the square, the
number of darts that hit the shaded part (circle quadrant) is proportional to the area of
that part.
Formula used to calculate the value of pi
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9. Fig.2 : Monte Carlo method applied to approximating the value of π. After placing 30000
random points, the estimate for π is within 0.07% of the actual value. This happens with an
approximate probability of 20%.
Explanation :
If each dart thrown lands somewhere inside the square, the ratio of "hits" (in the shaded
area) to "throws" will be one-fourth the value of pi. If you actually do this experiment,
you'll soon realize that it takes a very large number of throws to get a decent value of
pi...well over 1,000. To make things easy on ourselves, we can have computers
generate random numbers.
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10. Simulation Using MatLAB Code
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11. Estimated values of pi for different values of n by applying the monte-carlo algorithm.
Tabulation of different value of pi for different values of n while applying
Monte-Carlo algorithm and calculation of approximate error.
S.No. Number of throws, n Value of pi Approximate
error
1
1000 3.1680 .028
2
5000 3.1200 .020
3
10000 3.1084 .0336
4
20000 3.1354 .0066
5
25000 3.1389 .0031
6
77000 3.1440 -.002
7
100000 3.1488 -.068
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12. APPLICATIONS OF 'MONTE-CARLO' METHOD
• PEREGRINE, a Monte Carlo all-particle transport code is developed for
calculating the dose deposition in patients receiving radiation therapy.
• Simulates the transport of neutrons, photons, electrons and protons through a
patient using a geometry derived from a computed tomography (CT) scan.
• It will help doctor to choose a treatment that maximized the radiation dose to the
tumor and minimizes the dose to normal tissues.
• Monte Carlo method is used because it is the most accurate way to simulate
radiation transport on a computer.
• Quantum Monte Carlo methods are powerful numerical approach to the
investigation of Quantum Many-Body system.
• The methods are also used in following processes -
 Random Number Generations
 Nuclear Reactor Design
 Traffic Flow
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13.  Dow-Jones forecasting
 Oil-Well explorations
 Economics
 Environmental – air pollution
 Biological – Thermodynamics of Aging
Benefits and drawback of Monte -Carlo
Method
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14. ADVANTAGES
• Very flexible. There is virtually no
limit to the analysis. Empirical
distributions can be handled.
• Can generally be easily extended
and developed as required.
• Easily understood by non-
mathematicians.
• Enables study of interactions
between components.
• Enables the inclusion of real-world
complications.
• Can be used to analyze large and
complex real-world situations.
DISADVANTAGES
• Usually requires a computer.
• Calculations can take much
longer than analytical models.
• Solutions are not exact, but
depend on the number of
repeated runs used to produce
the output statistics
RESULTS OBTAINED
• Study on Monte-Carlo Method is done.
• Simulation was done in Matlab software for Specific Problem: Calculation of pi
Using Matlab through Monte Carlo method. Tabulation for different values of
throw was also done.
• Study on various application of Monte-Carlo Method and on its area was done.
• Step was studied for defining computational algorithm.
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15. REFERENCES
1. www. wikipedia.org/wiki/Monte_Carlo_method
2. www.mathswork.in
3. Modeling Derivatives using C++, Justin London, 7 edition, Springer
Publications
4. Financial derivatives, Journal of Derivatives, Volume 26, Pages 111-119,
2012.
5. Engineering Optimization: Theory and Practice, Fourth Edition Singiresu S.
Rao Copyright © 2009 by John Wiley & Sons, Inc
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