Monte Carlo and Markov Chain
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The document discusses Monte Carlo simulation, which is a method for estimating unknown quantities using statistical principles. It involves randomly sampling input variables based on their probability distributions, running simulations multiple times, and analyzing the results. For example, a Monte Carlo simulation could randomly generate task durations hundreds or thousands of times to estimate the probability of completing a project within a certain time frame. The key benefits of Monte Carlo simulation are that it allows estimating densities, means, and variances of distributions, as well as optimizing functions.
Monte Carlo and Markov Chain
- 1. Probability and Statistics Week 6 – Monte Carlo Simulation Dr. Ferdin Joe John Joseph
- 2. Monte Carlo Simulation • A method of estimating the value of an unknown quantity using the principles of inferential statistics • Inferential statistics • Population: a set of examples • Sample: a proper subset of a population • Key fact: a random sample tends to exhibit the same properties as the population from which it is drawn 2
- 3. Uncertainty • When you develop a forecasting model • any model that plans ahead for the future • you make certain assumptions. These might be assumptions about the investment return on a portfolio, the cost of a construction project, or how long it will take to complete a certain task. Because these are projections into the future, the best you can do is estimate the expected value. 3
- 4. How it works • In a Monte Carlo simulation, a random value is selected for each of the tasks, based on the range of estimates. • The model is calculated based on this random value. The result of the model is recorded, and the process is repeated. A typical Monte Carlo simulation calculates the model hundreds or thousands of times, each time using different randomly-selected values. 4
- 5. Example 5
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- 7. Solution scenario • In the Monte Carlo simulation, we will randomly generate values for each of the tasks, then calculate the total time to completion. The simulation will be run 500 times. Based on the results of the simulation, we will be able to describe some of the characteristics of the risk in the model. • To test the likelihood of a particular result, we count how many times the model returned that result in the simulation. In this case, we want to know how many times the result was less than or equal to a particular number of months. 7
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- 10. Why Monte Carlo • There are three main reasons to use Monte Carlo methods to randomly sample a probability distribution. They are: • Estimate density, gather samples to approximate the distribution of a target function. • Approximate a quantity, such as the mean or variance of a distribution. • Optimize a function, locate a sample that maximizes or minimizes the target function. 10
- 11. Sampling in Monte Carlo • Direct Sampling. Sampling the distribution directly without prior information. • Importance Sampling. Sampling from a simpler approximation of the target distribution. • Rejection Sampling. Sampling from a broader distribution and only considering samples within a region of the sampled distribution. 11
- 12. Applications Monte Carlo methods can be used for: • Calculating the probability of a move by an opponent in a complex game. • Calculating the probability of a weather event in the future. • Calculating the probability of a vehicle crash under specific conditions. 12
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