Dear Digby Millikan, William Harper told us that the area under the Gauss density f(x)=exp(-x^2 /2)/(sqrt(2pi)) is 1 and we all know that he is right.
Anyway I'll try to answer your consern, how that could be true. Indeed to those who might smile on Digbys consern, I have to say that this is a nontrivial finding of higher mathematical calculus, which probably non of the non mathematicians in the mailing list could prove. On one hand you need Lebesgue integration theory or at least improper Riemann integrategrals to even define, what the area under an infinite curve might be. Second the actual value is to my knowledge most easy obtained as a result of the residual theorem from function theory. Or has anyone a simple idea where the pi comes in and don't forget that the integral function of the curve does not have a closed form. Its just that we learned that the area is one, and we are good belivers. > I was wondering how the area under a gaussian distribution curve is 1 if > the tails go off to infinite?, the area must be infinite unless some unless some Your concern might be reduced to the concern, how a sum of infinitly many summands might stay finite, since the area under the tails is the sum of the areas in the intervalls from i to i+1 for all integer number i. I will give a simple example of such sum of infinitly many summands adding to 1. The i-th summand should be a_i = 0.5^i. We start to add 0 +0.5 = 0.5 0.5 +0.25 = 0.75 0.75 +0.125 = 0.875 ... In each line we added the half of that what was missing to 1 0 +0.5 = 0.5 ; (1-0 )/2=0.5 0.5 +0.25 = 0.75 ; (1-0.5 )/2=0.25 0.75 +0.125 = 0.875 ; (1-0.75)/2=0.125 ... It is easy to accept that we never get a value bigger than 1 because we always only add the half of what was missing to 1. On the other hand the distance to 1 is cut down to a half in every step and thus finally the distance to 1 drops under any e > 0. The abstraction that in this case the values of the sequence converges to 1 and that than the value of the infinite sum is 1, is one of the great and early historic achievments of calculus long before derivatives and integration and is linked to the initial definition of real numbers (which are actually defined as equivalence classes of (Cauchy)-sequences of rational numbers). A historic joke on that problem of finite sequences is the saying about the greek hero Achill who was known as a fast runner and a turtel. It goes as follows: A turle is running (or better say slowly crawling) 10 meter in front of Achill. Can he catch the turtle. The historic argument was: No, because in the time Achill needs to run the 10meter (say 1sec, he was really fast) the turtle advanced a little say 1 Meter, than in the time Achill needs to run this meter the turle advances again, and so on for ever again, such that Achill never reaches the turtle, since we get an infinite sum of timeslices before he reaches the turtle. However all of us know that after 2 seconds Achill has passed the turtle by 8 Meter since he got 20 Meters and the turle only 2. So we all know that Achill catches the turle. Only our brains play a trick on us in making us believe that an infinite sum of times must be infinite hindering Achill to catch the turle by taking him infinite time to run after it. Merry Chrismas, Gerald v.d. Boogaart Am Dienstag, 20. Dezember 2005 00:03 schrieb Digby Millikan: > Dear Madam/Sir, > > > > I was wondering how the area under a gaussian distribution curve is 1 if > the tails > > go off to infinite?, the area must be infinite unless some unless some > approximation > > is made? > > > > Digby -- ------------------------------------------------- Prof. Dr. K. Gerald v.d. Boogaart Professor als Juniorprofessor fuer Statistik http://www.math-inf.uni-greifswald.de/statistik/ office: Franz-Mehring-Str. 48, 1.Etage rechts e-mail: [EMAIL PROTECTED] phone: 00+49 (0)3834/86-4621 fax: 00+49 (0)89-1488-293932 (Faxmail) fax: 00+49 (0)3834/86-4615 (Institut) paper-mail: Ernst-Moritz-Arndt-Universitaet Greifswald Institut f�r Mathematik und Informatik Jahnstr. 15a 17487 Greifswald Germany --------------------------------------------------
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