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.

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
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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.
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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.
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5. Example
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6. Solution 1
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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.
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8. Solution
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9. Exponential Distribution
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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.
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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.
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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.
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13. Source code
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14. Monte Carlo for dummies
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15. Monte Carlo to dummies
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16. Monte Carlo to dummies
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17. Monte Carlo to dummies
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18. Monte Carlo to dummies
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19. Monte Carlo to dummies
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20. Monte Carlo to dummies
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