Quantitative Risk - Probability distribution

Continuous Distributions 

In the previous chapter, we learnt the basic details about Continuous Distributions. In this chapter, we are going to take a detailed look at some of the continuous distributions that are part of the PMI RMP Examination Syllabus.

There are many other types of continuous distributions. Before we jump into the details about each of these, let’s look at some general points about them.

• Continuous Distributions are typically used for cost, time and quality metrics
• They show Uncertainty in values
• The values shown in a continuous distribution are infinitely divisible (time, mass, distance etc.)
• They may have a probability of 0 as well

If something is infinitely divisible (like time which can be in years, months, weeks, days, hours, minutes, seconds etc.) you can probably express that as a continuous distribution

Some Important Points to Remember – About Continuous Distributions: 

• Beta and Triangular are the most common types of distributions used in Quantitative Risk Analysis
• Three-Point estimates are displayed using Triangular distributions. They can also be displayed using beta distributions
• Standard Deviations are displayed using normal or lognormal distributions
• Modeling and Simulation frequently use continuous distributions

Trivia:

In real life you may feel that some other types of distributions are widely used in quantitative risk analysis but from the exam perspective we will consider that beta and triangular are the most commonly used distributions because the PMBOK says so... 

Types of Continuous Distributions:

Actually speaking, there are many different types of continuous distributions. For the RMP Exam we will need to know about:

1. Beta
2. Triangular
3. Uniform
4. Normal
5. Lognormal and
6. Cumulative 

We will also be covering the definitions of Exponential and Gamma distributions for the same of completeness. However, those two are not part of the RMP Exam syllabus.

Beta Distribution: 

The Beta Distribution
• Is used to describe the uncertainty about the probability of occurrence of an event
• Is based on two shaped parameters
• Uses a range from 0 to 1 and can take several types of shapes.

The following is a sample Beta Distribution:




Triangular Distribution: 

The Triangular Distribution:
• Uses the estimate values based on the 3 point estimates that we covered during the chapter on Interviewing. The Optimistic, Most Likely and Pessimistic values from the 3 point estimate will be used here
• Use only 3 values
• Is used to quantify risks for each of the WBS elements

Remember WBS? WBS stands for Work Breakdown Structure.

The following is a sample Triangular Distribution

Quantitative Risk - Probability distribution

Probability distribution

Typically, a project's qualitative risk assessment will recognize some risks whose occurrence is so likely or whose consequences are so serious that further quantitative analysis is warranted. A key purpose of quantitative risk analysis is to combine the effects of the various identified and assessed risk events into an overall project risk estimate. More commonly, the overall risk assessment is used to determine cost and schedule contingency values and to quantify individual impacts of high-risk events. The ultimate purpose of quantitative analysis, however, is not only to compute numerical risk values but also to provide a basis for evaluating the effectiveness of risk management or risk allocation strategies.

The most stringent methods are those that require as inputs probability distributions for the various performance, schedule, and costs risks. Risk variables are differentiated based on whether they can take on any value in a range (continuous variables) or whether they can assume only certain distinct values (discrete variables). Whether a risk variable is discrete or continuous, two other considerations are important in defining an input probability: its central tendency and its range or dispersion. An input variable's mean and mode are alternative measures of central tendency; the mode is the most likely value across the variable's range. The mean is the value when the variable has a 50 percent chance of taking on a value that is greater and a 50 percent chance of taking a value that is lower. The mode and the mean of two examples of continuous distributions are illustrated in figure 14.

Figure 14: Mean and mode in normal and lognormal distributions

 

Quantitative risk analysis simulation starts with the model of the project and either its projectschedule or its cost estimate, depending on the objective. The degree of uncertainty in eachschedule activity and each line‐item cost element is represented by a probability distribution.The probability distribution is usually specified by determining the optimistic, the mostlikely, and the pessimistic values for the activity or cost element. This is typically called the“3‐point estimate.” The three points are estimated by the project team or other subject matterexperts who focus on the schedule or cost elements one at a time.

Specialized simulation software runs (iterates) the project schedule or cost estimate modelmany times, drawing duration or cost values for each iteration at random from the probabilitydistribution derived from the 3‐point estimates for each element. The software produces aprobability distribution of possible completion dates and project costs. From this distribution,it is possible to answer such questions as:

 How likely is the current plan to come in on schedule or on budget?

 How much contingency reserve of time or money is needed to provide a sufficientdegree of confidence?

Which activities or line‐item cost elements contribute the most to the possibility ofoverrunning schedule or cost targets can be determined by performing sensitivity analysiswith the software.

To select the correct probability distribution:

1. Look at the variable in question. List everything you know about the conditions surrounding this variable.
   You might be able to gather valuable information about the uncertain variable from historical data. If historical data are not available, use your own judgment, based on experience, to list everything you know about the uncertain variable.
   For example, look at the variable "patients cured" that is discussed in Tutorial 2 — Vision Research. The company plans to test 100 patients. You know that the patients will either be cured or not cured. And, you know that the drug has shown a cure rate of around 0.25 (25%). These facts are the conditions surrounding the variable.
2. Review the descriptions of the probability distributions.
   Probability Distribution Descriptions describes each distribution in detail, outlining the conditions underlying the distribution and providing real-world examples of each distribution type. As you review the descriptions, look for a distribution that features the conditions you have listed for this variable.
3. Select the distribution that characterizes this variable.