Poisson Distribution

Explanation

We tend to envision a startup business: a novice Filomena and her first few jars of frog legs, up against hardbitten and uncultured manager Filbert. But this is not what the problem implies. We were told that the rate of sale of frog legs at Filbert's was r = 5. That can only mean that previous sales experience existed, from which to calculate that average. It is also very likely that in some weeks more than 5 jars were sold, or the average of 5 per week could never have been reached. It follows that, in a typical week during the period when this average was established, more than 5 jars were available for sale. There is then no question of more frog leg fanciers being discovered, if contact with the idea of frog legs were only more present to the customer mind. We have proved that the idea was amply present to the customer mind, and it turns out that the customer palate has stabilized at an average of 5 jars per week as its considered verdict in the matter of frog legs.

The Fortune Magazine statistical oracle says that to avoid disappointing converted or speculative frog leg buyers, Filbert should start off with 9 jars. Very good, that is correct. But what about Week 2? By established fact (and if there is no established constant rate r, there is no Poisson problem at all), he will most likely sell 5 of those jars and end up with 4 left over. These, plus a next Monday input of 5 new jars, will give him once more 9 jars to open the week with, and the same calculation applies all over again. Some weeks he will sell a few more, some weeks a few less, but the restabilizing model will apply in general. So to Fortune's "9 jars to start," Filbert is simply saying, "Yes, but 5 jars a week to maintain."

Filbert is ultimately correct. It doesn't matter that we can overhear Filomena, in Aisle 11, saying to the indifferent cartpushers, "If you would only try them." By the terms of the problem, she is wasting her time, since the rate of sale has become established for that store and its clientele. Filomena can move to another frog swamp, near a tonier town, or she can start a mail-order business, but insofar as her public can be reached through the shelves of Filbert's Finer Feedables, Filomena has topped out.

Comment

So Fortune's advice to buy 9 rather than 5 jars per week, though it nearly double's Filomena's income, will only lead Filbert into excess inventory and lost liquidity. Someone in the back row will yell, "Do you mean to imply that Fortune Magazine favors the interest of manufacturer Filomena over that of middleman Filbert?

Hey, you said it, not us.