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Estimating Probability Distributions: Part 1 [Business Models
Posted on July 11, 2013 @ 07:56:00 AM by Paul Meagher

In my previous blogs on modelling revenue for a season of lobster fishing I was fortunate in having data I could work with that allowed me to specify detailed probability distributions for the main revenue factors in my revenue model. I modelled the distribution of catch sizes with a normal distribution and the distribution of prices with a categorical distribution (I knew what the 4 price points were and roughly what their relative probabilities were). I was able to make some fairly strong assumptions about how the revenue factors (catch size in lbs and price per lb) in my revenue model were probabilistically distributed.

When a startup is trying to model their expected revenues for a forecast period, there is often more uncertainty regarding how the relevant factors in their revenue model might be distributed (these "relevant factors" can also be called "random variables"). In such cases, we may need to resort to modelling these random variables (e.g., monthly sales) with distributions that are easier to specify and take better account of our level of uncertainty.

In this blog I want to discuss 2 distributions that are useful in such situations: a uniform distribution and a triangular distribution.

If you need to forecast the level of sales over a forecast period but are new to the market place and are uncertain as to what the uptake of your product or service will be; or are uncertain about the level of production that you might be able to achieve (e.g., crop yield using a new growing technique), then you might want to consider using a uniform distribution to represent your level of sales. Why a uniform distribution? To specify the parameters for a uniform distribution, all you need to specify are the upper and lower bounds of that distribution (denoted a and b in the graph below). You assume that the actual level of sales can fall anywhere within that range with equal probability (i.e., 1/(b-a)). Specifying the upper and lower bounds for your level of sales is significantly easier than specifying the expected mean and standard deviation for, say, your monthly sales figures. Also, it can be argued that a uniform distribution better reflects your more extreme state of uncertainty with respect to the variable you are trying to predict; namely, your level of sales for each month or quarter in your forecast period.

Figure 1: Uniform Probability Distribution

Source: http://en.wikipedia.org/wiki/Uniform_distribution_(continuous)

If you have a bit more confidence about what your most likely level of sales might be, and also what the upper and lower bounds of your sales might be, then you should consider using a triangular distribution to represent your uncertainty about your level of sales. To specify the parameters for a triangular distribution, all you need to specify are three values: the lower bound, the upper bound, and the most likely value (or modal value). These values are denoted as a, b, and c respectively in the graph below. The probability of your most likely value is computed using the formula 2/(b-a).

Figure 2: Triangular Probability Distribution

Source: http://en.wikipedia.org/wiki/Triangular_distribution

My library of probability distribution functions includes a UniformDistribution.php object and a TriangleDistribution.php object that could be used to generate random values from these distributions, after you specify the relevant parameters to them. This means that even under conditions of extreme uncertainty regarding expected sales, you may still be able to model expected revenue, and expected variance in revenue, if you opt to model your revenue factors using a uniform or a triangular probability distribution. In my lobster fishing example, if I was more uncertain about the catch size or catch price to expect, I might opt to use a uniform or a triangular distribution to model the distribution of possible catch sizes and catch prices rather than the normal distribution and categorical distributions I chose because I had more data to go on.

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