Working Pareto's distribution in Excel, I use the function P=F ^ [1/(1-F) ] to estimate the fraction P that receives more than one media. When F value approaches 1 from below, it results that P tends to P=0,36787926 just the inverse of "e" ... Is this known for you? How do you explain this result that surprised me?
Abdallah, B. Chandrasekar and Alejandro: I would welcome your comments too. It is good to confirm that creativity and intuition is esential in mathematical world, as also to pay attention to the contributions of mathematical granparents of centuries and milleniums ago. The next step is the interpretation of number "e". While pi=3.1416 has a visual and geometrical explanation, we can not say the same for "e". In this case its presence occurs when changing from a uniform Pareto distribution F=1 to a F=(1-delta) case, like a particular transition from monotony to diversity. Do we have good interpretations about "e" that help to explain its evident existence and application in practice? Thanks, emilio
Thanks Alejandro. You "set F=1+(1/n)" so F becomes bigger than unity. This is not possible in Pareto´s distribution. Does this change your analysis and how? Emilio
This second demonstration based in L'Hopital -which I need to revisit again after 40 years- looks nice and elegant, Alejandro. It satisfies my questions so I will conserve it. Thank you very much, emilio
The previous answers didn't let additional things to say but I'd just like to suggest an alternative method based on a change of variable and an asymptotic development. Please see in attach there. best.
a
Put F = 1 - 1/n so that n tends to infinity as F tends to 1. Now the result follows from the limit definition of e.
Abdallah, B. Chandrasekar and Alejandro: I would welcome your comments too. It is good to confirm that creativity and intuition is esential in mathematical world, as also to pay attention to the contributions of mathematical granparents of centuries and milleniums ago. The next step is the interpretation of number "e". While pi=3.1416 has a visual and geometrical explanation, we can not say the same for "e". In this case its presence occurs when changing from a uniform Pareto distribution F=1 to a F=(1-delta) case, like a particular transition from monotony to diversity. Do we have good interpretations about "e" that help to explain its evident existence and application in practice? Thanks, emilio
George, recently I found that if we graph the function Y(X)= x^(1/X) for real numbers bigger than zero, it happends that Y gests its maximum at x=e. This can be proved easily with maths, but the proof assumes that number "e" is well known and well applied. The nice point is that it may be also observed in the graph. OK, best wishes, emilio
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