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lectures_monte_carlo

# lectures_monte_carlo - Part II Random Processes Goals for...

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Part II - Random Processes Goals for this unit: Give overview of concepts from discrete probability Give analogous concepts from continuous probability See how the Monte Carlo method can be viewed as sampling technique See how Matlab can help us to simulate random processes See applications of the Monte Carlo method such as approximating an inte- gral or finding extrema of a function. See how Random Walks can be used to simulate experiments that can’t typically be done in any other way.

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Random processes are important because most real world problems exhibit random variations; the term stochastic processes is also used. The Monte Carlo Method is based on the principles of probability and statis- tics so we will first look at some basic ideas of probability. Similar to the situation where we looked at continuous and discrete problems (recall this was the main division in Algorithms I and II) we will consider concepts from both discrete and continuous probability. When we talk about discrete probability we consider experiments with a finite number of possible outcomes. For example, if we flip a coin or roll a die we have a fixed number of outcomes. When we talk about continuous probability we consider experiments where the random variable takes on all values in a range. For example, if we spin a spinner and see what point on the circle it lands, the random variable is the point and it takes on all values on the circle.
Historical Notes Archaeologists have found evidence of games of chance on prehistoric digs, show- ing that gaming and gambling have been a major pastime for different peoples since the dawn of civilization. Given the Greek, Egyptian, Chinese, and Indian dynasties’ other great mathematical discoveries (many of which predated the more often quoted European works) and the propensity of people to gamble, one would expect the mathematics of chance to have been one of the earliest devel- oped. Surprisingly, it wasn’t until the 17th century that a rigorous mathematics of probability was developed by French mathematicians Pierre de Fermat and Blaise Pascal. 1 1 Information taken from MathForum, http://mathforum.org/isaac/problems/prob1.html

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The problem that inspired the development of mathematical probability in Re- naissance Europe was the problem of points. It can be stated this way: Two equally skilled players are interrupted while playing a game of chance for a certain amount of money. Given the score of the game at that point, how should the stakes be divided? In this case “equally skilled” indicates that each player started the game with an equal chance of winning, for whatever reason. For the sake of illustration, imagine the following scenario.
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lectures_monte_carlo - Part II Random Processes Goals for...

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