An additional improvement we can make to the revenue model for our lobster fishing example involves a more realistic distribution of prices per lb for lobster. Previously we assumed that the distribution of prices was normally distributed around a mean of $3.50 a lb with a standard deviation of 25 cents per lb. This means that prices can range continuously around $3.50 with the probability of lower or higher prices falling off as we deviate from $3.50. A price as extreme $3.00 a lb is 2 standard deviations away from the mean price and is expected to occur for some catches when we represent lobster prices with a normal distribution with these parameter settings (e.g., mean=3.50, stdev=0.25).

The problem is that a price of $3.00 never occurred during the season. Also, the prices never varied continuously. Instead the prices only varied in 25 cent increments so there were really only 4 price points in our price distribution; namely, 3.25, 3.50, 3.75, and 4.00. Also, the lower prices in this range occurred more often than the higher prices. Using a normal distribution to capture the distribution of prices is a simple first approximation and may put us in the ballpark for the average prices over the season, but it does not reflect the true state of affairs with respect to what prices were possible and their probabilities of occurrence.

In order to construct a more realistic price distribution, we can opt to represent the distribution of prices with a categorical distribution. I also considered calling this distribution a discrete distribution or a multinomial distribution, but these terms carry a bit of extra baggage that I did not want to commit to at this time.

A categorical distribution consists of labels along with the associated probability of each label. Collectively the probabilities of all labels should sum to 1. The main problem I want to solve in today's blog is how to generate a random label from this distribution. My approach involves constructing a new distribution called CategoricalDistribution.php and developing a random number generator for that distribution.

Here is what a "bare bones" version of a CategoricalDistribution.php object looks like:

One important aspect of this code to take note of is that the distribution includes a general ProbabilityDistribution.php object (located in the Probability Distributions Library that I developed). This object contains a random number generator function, called RNG(), that calls the private _getRNG() method in this code. The parent RNG() method (in the ProbabilityDistribution.php object) can call this private method multiple times if it is supplied with a number specifying how many times to call the private method; otherwise, it just returns one random category label. The CategoricalDistribution.php object is "bare bones" because it does not include a variety of other methods/functions that are usually included in probability distribution objects, such a methods for returning the mean, standard deviation, pdf values, cdf values, etc... for a distribution. Perhaps in the future I will add these additional methods after I have reviewed the literature, but for current purposes I only needed to implement the private _getRNG() method (private methods are often prefixed with an underscore to make their status as private methods more obvious).

To verify that the random variate generator for categorically distributed variables works as it should, I have developed a random_prices_test.php script that is supplied with a distribution of price points and their corresponding probabilities (i.e., $price_distribution = array('3.25'=>0.4, '3.50'=>0.3, '3.75'=>0.2, '4.00'=>0.1)). It generates 30 random prices and then outputs this array of 30 random prices.

Here is what the output of this script looks like:

Array
(
    [0] => 3.50
    [1] => 3.50
    [2] => 3.75
    [3] => 3.25
    [4] => 3.50
    [5] => 3.75
    [6] => 3.25
    [7] => 4.00
    [8] => 3.75
    [9] => 3.25
    [10] => 3.50
    [11] => 3.75
    [12] => 3.50
    [13] => 3.25
    [14] => 3.25
    [15] => 3.75
    [16] => 3.25
    [17] => 3.50
    [18] => 3.25
    [19] => 3.50
    [20] => 3.50
    [21] => 3.75
    [22] => 3.25
    [23] => 3.50
    [24] => 3.25
    [25] => 3.25
    [26] => 3.75
    [27] => 3.50
    [28] => 4.00
    [29] => 4.00
)

Notice how the value 3.25 occurs more than other values and that a value like 4.00 occurs the least often. The random variate generator appears to be working as it should (more sophisticated tests than the eyeball method are possible but I'll spare you the details). We can use the CategoricalDistribution.php object in our next blog to construct a more realistic model of lobster fishing revenue.

Conclusion

In today's blog I introduced you to the idea of a categorical probability distribution because it is required in order to develop a more realistic model of lobster fishing revenue (i.e., lobster price points are distributed in a categorical manner). In general, when modelling revenue from a line-of-business you not only need to give careful consideration to whether the factors determining revenue are stationary or non-stationary (which I discussed in my last blog), but also whether the factor values are distributed in a continuous (e.g., lobster catch size) or a discrete manner (e.g., lobster prices). If you keep these two critical distinctions in mind then you should be able to generate much more realistic revenue models for your own line-of-business.

From a lobster buyer's point of view the price they are willing to pay for lobsters varies according to supply and demand aspects of their business. If we knew what these factors were, we might be able to develop a non-stationary representation of lobster prices that take these factors into account. Because sellers are often in the dark about the buyer's business, we must often be content to use a stationary price distribution to reflect our uncertainty regarding the price we can sell our goods for.

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