This is the sixth post about the Companion and Self-Testing Question List for my recent book Business Risk and Simulation Modelling in Practice: Using Excel, VBA and @RISK. As mentioned in earlier posts, the complete list of questions posted in this series will also shortly be available for free on the John Wiley web-site.

As a reminder, the full series of posts correspond to the Chapters of the book as follows:

- Post I: Risk Assessment Context and Processes (Chapters 1 and 2 of book)
- Post II: Risk Assessment, Quantification and Modelling: Approaches, Benefits and Challenges (Chapters 3, 4 and 5)
- Post III: Principles of Simulation Methods (Chapter 6)
- Post IV: Core Principles in the Design of Risk Models (Chapter 7)
- Post V: Measuring Risk using Statistics of Distributions (Chapter 8)
- Post VI: The Selection of Distributions for Use in Risk Models (Chapters 9 and 10)
- Post VII: Modelling Dependencies between Sources of Risk (Chapter 11)
- Post VIII: Using Excel/VBA for Simulation Modelling (Chapter 12)
- Post IX: Using @RISK for Simulation Modelling (Chapter 13)

This is Post VI, The Selection of Distributions for Use in Risk Models (Chapters 9 and 10):

- What is the role of the uniform continuous distribution in risk modelling?

- What are the main advantages and disadvantages of using a uniform continuous distribution to capture uncertainty?
- What is meant by a Bernoulli distribution? What is the difference to a binomial distribution?
- What are the main reasons to use a triangular distribution, and what are the main disadvantages?
- How does a normal distribution arise?
- What are some ways to use a triangular distribution to approximate a normal distribution?
- How does a lognormal distribution arise?
- What is meant by the natural and the logarithmic parameters of a lognormal distribution?
- Show the mathematical steps to convert from natural to logarithmic parameters, and vice versa (Additional exercises: implement these in Excel).
- What is the equation for the mode of a lognormal distribution in terms of its natural and its logarithmic parameters? (Additional exercise: using @RISK, create some examples of this with the accompanying graphs)
- How does a beta distribution arise?
- Explain how estimates of a probability of occurrence that are derived from small sample sizes may be inaccurate.
- How can a PERT distribution be created from a beta general distribution?
- What are the key similarities and differences between a triangular and a PERT distribution?
- What does the Poisson distribution describe?
- Give several examples of processes that could be reasonably expected to be Poisson distributed, and exampling why one may expect this.
- Which distributions approximate a low- and a high-intensity Poisson process respectively?
- How does a geometric distribution arise? What is the relationship to the Exponential distribution, and to the negative binomial distribution?
- What are some key uses of the Weibull distribution?
- What are some uses of the gamma distribution?
- Describe the general discrete distribution and the integer uniform distribution, and explain how they may be used in some risk modelling applications.
- What process is described by the hypergeometric distribution?
- What are some key uses of the Pareto and the extreme value distributions?
- What are the similarities and differences between a normal distribution and a logistic distribution?
- What are the main properties and uses of a Student (T) distribution? Explain how the various implementations of the distribution functions in Excel relate to each other.
- Which distribution arises as a result of adding many identical and independent variables?
- Which distribution arises as a result of multiplying many identical and independent variables?
- Which distribution corresponds to the time to occurrence for a binomial process?
- Give three examples of processes which are likely to be Poisson distributed.
- What are the main categories of methods to use when considering the appropriate distribution to use in a risk model?
- What is meant by the alternative parameter form of a distribution (in @RISK) and what are the main benefits of using this approach?
- Show how the values of the standard parameters of a Weibull distribution can be derived from any two percentiles (e.g. the P10 and P90).
- Describe how (in @RISK) the RiskTheo and other functionality may be used to try to approximate one distribution with another.
- What probabilities (weights) must be given to the values of the P10, P50, P90 of a Normal distribution so that a General Discrete distribution using these percentiles and probabilities has the same mean and standard deviation as the original Normal distribution?
- When using the alternate parameter form of a PERT distribution, what are the advantages and disadvantages of using (as an input parameter) the most likely (modal) versus using the P50 value?
- What happens to the skewness of a distribution, created using alternate parameters, as the percentages used for the percentiles increase e.g. when using P1/P99 in place of the P5/P95 or in place of the P10/P90)? What practical tools are contained within @RISK to determine this?

- What are some frequent pitfalls when using information from a risk register to try to build full risk models?
- What is the difference between the definition of distribution functions in Excel and in @RISK?
- Which distributions in Excel have readily-available functions to allow random sampling to be directly conducted?
- Using mathematical manipulation, starting from the formulae for the density or cumulative curves, derive the equations for the inverse (percentile) functions that are given in Section 10.1.2.
- What techniques can be used when random samples cannot be calculated directly from a probability, but the cumulative distribution function is known?
- What are the advantages of using user-defined functions to create random samples from various distributions in Excel?
- Write the appropriate VBA code to create random samples for several of the distributions given in Chapter 10 of the book.

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