Part 2. MM4XL Tools > 1. Strategic Tools > Risk Analyst > 2. Simulation Never heard of it > What are probability distribution functions?

Risk Analyst

What are probability distribution functions?

Probability distribution functions (Pdfs) are statistical devices that marketers can use to model business assumptions. For instance, a common business assumption concerns market share; the statement next year our market share will be in the range 3.5%-4.5%, most likely 3.9% is equal to saying next year our market share will be distributed triangularly within Min = 3.5%, Max = 4.5%, and ML = 3.9%. Also, next year market size will be between 85 and 115 Mio. is equivalent to saying next year market size will be normally shaped with mean 100 Mio. and standard deviation 5 Mio.

Scenario modelers have found that old-style models built using single-bullet variables too often do not represent an acceptable model of real events. For this reason, the old static modeling fashion has lost ground in favor of dynamic modeling. With dymanic modeling, instead of inputting a fixed value for an uncertain variable, say 5% for our future market share, more sophisticated models are built using variables defined within a range of values. When the dynamic model is repeated many times (for example, 1000 times), and each time a new value within the boundaries of the distribution is used for the uncertain variable, we can collect 1000 different simulated observations of our market share. When modeled well, a distribution has a higher probability of including the real value of the uncertain variable than a single-bullet figure has.

There can be as many different kinds of assumptions as there are Pdfs, and this may cause some trouble for new users. However, there are many advantages to building models on assumptions defined with Pdfs, and they may justify the moderate learning effort required for applying such devices. It must also be said you do not always need to use spectacularly complex Pdfs to model assumptions. Many useful models are based on fairly simple assumptions.

When working with Risk Analyst, values within boundaries can be produced for many different distributions. We will begin by using perhaps the simplest of these functions:

=mmRANDBETWEEN(4%, 6%)

Copying the formula above in 1000 cells and summarizing the results in, say, 15 classes, yields a chart like the one below (we used the function mmHISTO to summarize the 1000 trials). In Class 1, 69 simulated values between 4% and 4.14% have been aggregated; Class 2 contains 55 values going from above 4.14% to 4.27%, and so on.

 Monte Carlo Simulation Software: Management Process Risk Analysis

This chart tells us that the 1000 runs were distributed in more or less equal shares across the 15 classes (more simulation trials would smooth out the differences). That is, each number in the range 4%-6% had equal likelihood of being chosen. This assumption of equal probability, however, is not always a reasonable one because it does not permit spreading the risk across the possible outcomes of a distribution.

An analyst knowledgeable of their market could assume that the most extreme values close to the tails of the distribution above may have a lower likelihood of occurring. In this case, the 4%-6% range could still be used but it should also be specified that a more likely value may occur, for instance, in the 5% range. Repeating the 1000 runs with the formula below produces a new distribution of triangular values:

=mmTRI(4%, 5%, 6%)

The chart below summarizes the 1000 triangular trials in 10 classes, and was made with the mmHISTO function.

 Monte Carlo Simulation Software: Management Process Risk Analysis

There is a difference between the two pictures above. In managerial issues this difference is relevant because it allows you to model the amount of risk linked to an event, and there is a lot of risk in management. For this reason too, there are many distributions available, each used for modeling one or more instances.

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