Lecture 6 Flashcards
Explain what a Monte Carlo simulation is
Monte Carlo simulation is used for numeric calculations. The output of a Monte Carlo simulation is typically a probability density function (PFD).
A cost estimation method will provide a point estimate, i.e. a specific number. A point estimation is typically insufficient for decision-making. Additionally, we would like to know the uncertainty of the point estimate. The uncertainty of the point estimate can be described with a statistical distribution. To identiy the statistical distribution of the investment cost, we can use Monte Carlo simulation.
When to use Monte Carlo simulation: ised used when we know the behavior of each input variable and their dependencies, but do not know (or cannot easily figure out) the behavior of the output variable.
What is the difference between the analytical and numeric approach?
- Analytical approach: provides an exact number to the problem. Regardless of who you are, everyone will end up with the same result, assuming the calculation is done correctly. For instance, 2 + 2 = 4 regardless of who you are. Analytical is the most rigorous approach and arguably the most well-respected. The shortcoming, however, is that it can be difficult or even impossible to conduct an analytical analysis on complex problems.
- Numeric approach: involves methods such as simulation. The method does not provide an exact answer and you could end up with a different answer every time you run the analysis. Numerical methods can often be easier to implement than the analytical method.
Explain what a pseudo-random number generator (PRNG) is.
- PRNG is the secret to stochastic variables in Monte Carlo simulations.
- A PRNG only generates uniform, continuous numbers between 0 and 1. A PRNG is a deterministic algorithm that produces numbers that satisfy certain criteria for randomness.
Characteristics:
* Produce numbers that are approximately uniform
* Produce numbers that are approximately independent in a moderate number of dimensions
* Have a sufficiently long period
* Be reproducible from a simply-specified starting point, but unless a starting point is specified, should be unpredictable.
In Excel:
* Excel worksheet: RAND()
* VBA: RND
* BOTH: will generate numbers between 0 and 1, with up to 15 decimals of precision.
What is the difference between PDF, CDF and QF?
- PDF (Probability Density Function): describes the likelihood of a continous random variable taking on a specific value.
- CDF (Cumulative Distribution Function): gives the probability of the variable being less than or equal to a certain value.
- QF (Quantile Function): Allows to find the value of the random variable corresponding to a specified probability.
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Explain what the universality of the uniform is.
The universality of the uniform allows for a straightforward and general approach to generating random variables with various distributions. By leveraging the QF of a distribution, one can transform uniformly distributed random numbers into random numbers that follow the desired distribution. This process underpins many statistical and computational techniques, making it a fundamental concept in probability theory and simulation.
Easier explaination: The universality of the uniform means that you can use random numbers between 0 and 1 to create any other type of random number you want by following a specific recipe called the quantile function. This lets you generate numbers that follow different patterns or distributions, like bell curves or exponential curves, just by knowing how to mix the original random numbers in the right way. It’s like having a versatile tool that turns simple randomness into complex patterns effortlessly.
Explain how do carry out a Kolmogorov-Smirnov test.
Kolmogorov-Smirnov test: finds the maximal vertical distance between the empirical cumulative distribution function (CDF) and the theoretical cumulative distribution function. If the vertical distance is sufficiently small, the test concludes the two distributions are coinciding.
Explain how to determine the probability of a project cost being in a certain interval.
If the distribution of project costs is known, you can use either the PDF or CDF of the distribution.
- Using the PDF: integrate it over the interval of interest to find the probability. Mathematically, it would be the integral of the PDF function over the specified interval. (You would integrate f(X) from a to b)
- Using the CDF: you can substract the CDF values at the enpoints of the interval. Mathematically, it would be P(b) - P(a), where P(x) is the CDF.
By using the appropriate mathematical tool (PDF or CDF) and the known distribution of project costs, you can calculate the probability of the project cost falling within any specified interval.
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Explain how we can simulate discrete uniform numbers.
Simulating discrete uniform numbers involves generating random integers from a specified range where each integer in the range has an equal probability of being selected.
Explain how we can simulate a discrete distribution.
Explain how we can simulate two correlated normally distributed variables.
Explain how we may simulate the cost of many correlated, work packages.
Be able to explain how to carry out a Monte Carlo simulation on CAPEX.
Explain how to make a histogram
