# Risk triangular distribution: percentile sensitivity

When modelling risks, usually a triangular distribution based on three time estimates (optimistic, likehood and pessimistic time estimates - a, c, and b respectively) is used to evaluate the planning task duration.

Once task distributions are known, a Monte Carlo algorithm generates an array of possible outcomes, resulting in better schedule overview when compared with a fully deterministic approach.

Because the triangular Probability Density Function (PDF) is zero in a and b, an appealing method to solve for a and b is based on using percentile estimates. As a matter of fact, it is possible to define a triangular PDF by specifying a lower percentile ap such that a < ap <= c <= br < b (see here: percentile). This approach avoids to specify the lower and upper extremes a and b that by definition have a zero likelihood of occurrence.

The value of p = 0.1 (then q = 1 - 0.1 = 0.9) specified in percentile is one of the possible value we can set for p. For instance I could choose p = 0.15 (q = 0.85). The point is that this value affect significantly the results of the Monte Carlo simulation, even when a large number of iteration (30000) is used for simulating risks. In other words, risk analysis result sensitivity to percentile is high.

So, is there a rule of thumb and/or mathematical constraints I have to account for to set the percentile value?

## 1 Answer

Mathematically, representing a triangular PDF as a/c/b values or as mode/percentiles is equivalent (you can convert each representation into the other while keeping the effective probability distribution). If you see differences in the Monte Carlo outcomes they are most likely caused by the way the percentile-based random function is implemented, I'd suspect it's not equivalent to the a/c/b triangle based random function, so it's probably simply incorrect (different methods of evaluating mathematically equivalent functions should return identical results).

That said, the triangular PDF is always a very rough simplification of the actual probability distribution of time to complete a task. In reality, the time distribution of tasks isn't triangular, and the values for a, b, and c are pretty much arbitrary values based on your experience, so they might result in a triangular PDF that is more or less congruent with the actual probability density of the task being estimated. In particular, b might be problematic for non-standard tasks - when the task you estimate is impossible to complete, the probability of it being finished in any finite time is zero :-)

My personal gut feeling as a software developer (I'm not project manager) is that task time to complete for tasks that involve searching for a solution (such as debugging, or coming up with a solution for a non-standard requirement) may be better modeled as a log-normal distribution, which takes into account the possibility of not finding a solution within any limited amount of time. Of course, even in software development many tasks are of a repetitive and finite nature, and those may be well enough estimated using a triangular PDF based on experience.