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Decision making through intuition_3/3_Exponential distribution



 

Often we complicate easy decisions with assumptions that do not really affect our systems. “Easy decision” does not necessarily mean that the best solution is easy to find, it means that often the best reasoning process we can use is a simple one. In those situations we should accept that our best action is likely not to result in the best possible outcome, it is however the best we can do.


Let us imagine we want to predict the life of a machinery, tool or device like our laptop; let’s also agree on an average life-expectancy of about 3 years. The function describing the probability of failure of your laptop in time (life-expectancy) is unlikely to be either one of the two distributions already discussed in this series: the Power-Law and Normal distribution. As we can perceive from the picture below, the Normal distribution seems to concentrate too much the probability of failure around the mean value and it also implies symmetry without distinction between being months or years before the average value or after it. The Power-Law distribution (also the 80-20 rule) prefers a little too much many failures happening well before the expected value and few occurring far away in the tails; however, our device is unlikely to last on average only 1 or 2 years and then have few outliers lasting 30 years. Moreover, note that the mean and variance of the Power-Law distribution may be loose concepts while we still think many laptops will need to be replaced after about 3 years, verifying the average value.




So, for the life of a laptop we need something describing the probability of failure as being proportional to the elapsing time and where there is still good probability that the mean value will be verified. The probability function we are looking for is likely to describe the duration of many real-life phenomena:


· The time required to the person in line ahead of us at the post office to be serviced.

· The time after which the machine in the manufacturing plant fails and needs to be replaced.

· The time needed for the train to arrive at the station.


The function we are looking for is the exponential distribution and we show an example of it below (life-expectancy of the laptop equal to 3 years):


Despite similarities, the tail of the exponential distribution (constant^x) decays faster than the PowerLaw's (x^constant) not verifying similar outliers. In general, something can be associated with the exponential distribution if the probability of the event occurring -- or finishing or failing -- is proportional to the time elapsed (the proportionality constant would be the “rate” = 1 / average value). There is one important property of phenomena with duration described by this distribution and it is the lack of memory. Memorylessness means that regardless of where we arrive on the x-axis, the probability of proceeding further a certain amount is the same as the probability, starting from zero, of going forward that same amount.


Example:

Let’s say that our laptop, with an average life of 3 years, is still functioning after 4 years. The probability of the laptop functioning another year is the same as the probability that a new laptop gets through its first year of life. In a similar way, if we want to estimate the probable life left after the 4th year, we can simply estimate it will live another average life of 3 years. Trying to calculate some kind of conditional probability on the base of thoughts like “considering it is still functioning after 4 years… it is likely to fail very soon – or -- last very long’’ is likely not to result in an accurate result.


The lack of memory may be misinterpreted as making the distribution useless because it can be restarted at any point in time. That statement though is missing an important point: the property is conditional on the laptop reaching a specific point in time, in our example the 4th year. The exponential distribution is giving us very important information: we should not overcomplicate the situation by making too many assumptions given what has happened so far. Imagine we are at the post-office and the average servicing time per customer is 5 minutes, if servicing the person in front of us has already taken twice that amount -- 10 minutes -- we can only guess it will take other 5 minutes for our turn to come.


We can conclude this series by going back to the enduro-rider we have been following since the first post: what would be his best option in case the exponential distribution applied to his case? The rider could guess the average time required to the other rider to proceed – about 10 seconds -- knowing thought that if the other rider is still stuck after that moment, he cannot do much more than predict other 10 seconds. In that situation, assuming the two routes comparable in time and just few seconds apart, we should not blame Graham for any decision. We could consider him to be right with any options available to him: take immediately the alternative route or wait for the other rider to proceed. Rather than making many different assumptions that could take us far away from the expected value, the input from the exponential distribution still constitutes very actionable information.


In some way, the exponential distribution reminds us that simplicity is key.


Despite our simple discussion in this series of three main distributions like the Power-Law, Normal and Exponential, we hope it will encourage the reader to conduct personal research and apply possible findings to specific experiences and domains.


 

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