Monte Carlo Simulation - Engineering Assignment Help

Assignment Task

1 Learning outcomes
The purpose of this experiment is to develop, explore and test Monte Carlo techniques in simulating and finding solutions to real-life random processes. MATLAB will be used as the tool to do the tests of the experiment, but it is not the main learning outcome (i.e. the experiment is not about MATLAB).

2 Introduction
The Monte Carlo method is a numerical method of solving mathematical problems by the simulation of random variables. The name Monte Carlo was applied to a class of mathematical methods first by scientists working on the development of nuclear weapons in Los Alamos in the 1940s. The essence of the method is the invention of games of chance whose behaviour and outcome can be used to study some interesting phenomena. While there is no essential link to computers, the effectiveness of numerical or simulated gambling as a serious scientific pursuit is enormously enhanced by the availability of modern digital computers.
The term “Monte Carlo” refers to procedures in which quantities of interest are approximated by generating many random realisations of a stochastic process and averaging them in some way. In statistics, the quantities of interest are the distributions of estimators and test statistics, the size of a test statistic under the null hypothesis, or the power of a test statistic under some specified alternative hypothesis.
How can we use Monte Carlo techniques to find the sampling distribution of an estimator? In the real world, we usually observe just one sample of a certain size N, which will give us just one estimate. The Monte Carlo experiment is a lab situation, where we replicate the real world study many (R) times. Every time, we draw a different sample of size N from the original population. Thus, we can calculate the estimate many times and any estimate will be a bit different. The empirical distribution of these many estimates approximates the true of the estimator. A Monte Carlo experiment involves the following steps:
(1) Draw a (pseudo) random sample of size N for the stochastic elements of the stochastic model from their respective probability distribution functions.
(2) Assume values for the parts of the model or draw them from their respective distribution function.
(3) Calculate the parts of the statistical model.
(4) Calculate the value (e.g. the estimate) you are interested in.
(5) Replicate step (1) to (4) R times.
(6) Examine the empirical distribution of the R values.

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