文档介绍：Monte Carlo Simulation Monte Monte Carlo Simulation Carlo Simulation 中国海洋大学数学学院中国海洋大学数学学院施心慧施心慧 Contents Contents ? Idea of Monte Carlo Simulation ? Simulating Deterministic Behavior: ?? Area Under a Curve Area Under a Curve ( (曲边梯形的面积曲边梯形的面积) ) ?? Volume Under a Surface Volume Under a Surface ( (曲顶柱体的体积曲顶柱体的体积) ) ? Simulating Probabilistic Behavior: ?? Flip Flip (抛、掷) (抛、掷) of a of a Fair Fair (均匀的) (均匀的) Coin Coin ?? Roll of a Fair Roll of a Fair Die Die (骰子) (骰子) ?? Roll of an Unfair Die Roll of an Unfair Die Idea of Monte Carlo Simulation ? Simulate behavior or event by generating random numbers of some distributions ?? Axioms of Probability Axioms of Probability Axioms of Probability One of the keys to understand Monte Carlo simulation is the axioms of axioms of probability. probability. The probability of The probability of some some event event can be considered as the limit of the frequency of that event as the number of trail s approaches to infinity. Area Under a Curve Area Under a Curve ( (曲边梯形的面积曲边梯形的面积) ) Use Monte Carlo simulation Monte Carlo simulation to model a deterministic behavior, the area under a curve . Find an approximate value of the area. Suppose f(x) is a continuous function satisfying 0 ( ) , f x M a x b ? ? ?? The number M is some constant that bounds the function Area Under a Curve Area Under a Curve ( (曲边梯形的面积曲边梯形的面积) ) Select a point P(x,y) at random within the rectangular region: [ , ] [0, ] a b M ? How can we do this on puter? How can we do this on puter? To generate a random number pair (x,y) uniformly distributed uniformly distributed (均匀分布) in the region. Area Under a Curve Area Under a Curve ( (曲边梯形的面积曲边梯形的面积) ) If this procedure is repeated for extremely many times, then we get area ar under curve of rect ea angle ? Note: the number of simulation should be very large Area Under a Curve Area Under a Curve ( (曲边梯形的面积曲边梯形的面积) ) belowcur number of counted total n