Various Uses of Distributions in Risk Analysis (1 of 3)

The following Risk Analysis Tip is provided by Dr. H. Groenendaal (Huybert@risk-modeling.com) at Vose Consulting, and has been drawn from material in ModelAssist® for Crystal Ball®, the comprehensive risk analysis training, reference and template software. ModelAssist users can consult the ModelAssist-references (in the form of Mxxx) for additional information. To read more about ModelAssist and get a free download of the demo version, click here.

Introduction

The correct design and development of Crystal Ball models is highly dependent on the risk analyst’s understanding of the exact use (meaning) of distributions (Crystal Ball assumptions) that he or she uses. In our training courses and during our consulting work, we however unfortunately find that most people do not really know or understand the differences between the various uses of probability distributions. While this subject is certainly not always very easy to understand, it is imperative for the accurate design and development of your Crystal Ball models.

The goal of this Risk Analysis tip is to make you aware of the three different uses of distributions and give you a short introduction to "uncertainty distributions."

Three Uses of Distributions

Probability distributions (i.e. Crystal Ball assumptions) can represent any of the following:

• Uncertainty distributions - describing our uncertainty about some model parameter
• Frequency distributions - describing variability between individuals
• Probability distributions - describing randomness

For the risk analyst (Crystal Ball modeler), it is very important to understand and realize which mean each of the distributions has in his/her model because this will dictate the correct use of it (e.g., while you are able to multiply two uncertainty distributions, you can NOT multiply two probability distributions).

1. Uncertainty Distributions

An uncertainty distribution describes the observer’s level of knowledge about a certain fixed parameter. With ‘fixed’, we mean that there is one true value of this parameter, but we just don’t know it. For example, if you want to estimate the height of a building or the average income of a country, both quantities have a single value, but there may be some uncertainty when estimating them (i.e. measuring tools may not be 100% accurate, and often we only sample part of the population)

Six Sigma example: Imagine you are trying to estimate the proportion of products that fail to meet certain requirement. To do this, you are doing a small experiment and will test 100 products – 6 of them appear not to meet the requirements. What is the probability a part does not meet requirements?

If you answer 6%, you are not acknowledging that we actually do have uncertainty about this exact probability. Therefore, instead we should express this percentage as an uncertainty distribution. The accurate distribution to use for this is the Beta distribution (see figure below). In addition to showing the uncertainty distribution when we have tested 100 products (n = 100), we also show the uncertainty distribution if we would have tested 1,000 products and 60 were not meeting requirements. As you can see, the uncertainty distribution when testing more products is a lot narrower, indicating the value of making more observations.

Insurance example: In your Crystal Ball model, you need to model the rate of car accidents in a certain state. You know that during the last 3 years, you have seen 120, 103 and 112 accidents. What is the true rate of accidents per year, assuming that no major changes have occurred in the safety regulations in this state and that all accidents have actually be reported? If you answer is (120+103+112)/3, you also ignore that you have only observed three years, and there is therefore uncertainly about the true underlying rate of accidents per year. You can therefore use the following Gamma (Location = 0, Scale = 1/3, Shape = 120+103+112) distribution to represent this uncertainty, as show below.

Financial (credit risk) example: The use of the beta-distribution in a previous Risk Analysis tip (see http://www.crystalball.com/support/risktips/risktip-7.html) illustrates another uncertainty distribution – in that case, we were uncertain about the true default probability, and we represented this uncertainty with a Beta distribution.

Conclusion

Model parameters are fixed values that have to be estimated either by statistical inference from observations, as discussed in this section above, or by expert opinion. An uncertainty distribution then represents our ‘level of knowledge’ (or ‘level of ignorance’) for a certain parameter in your Crystal Ball model, and can be a realistic way of incorporating parameters into your models.

In future Risk Analysis tips, we will also cover the fact that we are usually uncertain about the parameters of the other two types of distributions (the frequency and probability distributions), and therefore can use uncertainty distributions to describe that uncertainty.

> View the next tip in this series

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* The material within this ‘Risk Analysis Tip’ comes from one of the over 500 risk analysis topics available in ModelAssist for Crystal Ball. ModelAssist for Crystal Ball gives a more detailed explanation of the above methods and any risk analysis techniques involved.