Dear members of the list, Is it the goal of the list to discuss mathematical basic calculus? Everything can be read e.g. in Abramowitz and Stegun (eds.) (1964): Handbook of Mathematical Functions. National Bureau of Standards.
Best regards, Rudi Dutter [EMAIL PROTECTED] wrote: > > Hello list, > > Here is the 'trick' I came across in diffusion studies, which might not not > fully satisfy the pure mathematicians in the list, but I can see the > picture, so I like it. This goes as follows :- > > Start with the standard statement that the integral from minus to plus > infinity of e raised to power -1/2((x-m)^2)/(s^2) dx = ssqrt(2p) > First step substitute y = (x-m)/s into the above relationship. > > Now we want to show that A= (is equivalent to) the transformed integral > from minus infinity to plus infinity of e raised to the power -1/2 y^2 dy = > sqrt(2p) > If we consider A^2, then we can write this as the product of two integrals > of the above form, one in x and the other in y. This is then combined as > the double integral (with the same limits) of e raised to the power -1 > /2(x^2 +y^2) dx dy. This double integral is the volume under the bell > shaped surface e^(-1/2(x^2 + y^2)). > > Now x^2 + y^2 = the square of the distance of point x, y from the origin, > denote this by r^2. The area of a ring, distance r from the origin, and > width dr, = 2prdr. > > The volume of a cylindrical shell with this annulus as its base, and with > height, e^(-1/2(r^2)), so that it just touches the bell shaped surface is > (e^(-1/2(r^2))) 2prdr > > Therefore A^2 = 2p times the integral from zero to infinity of (r times > e^(-1/2(r^2)) dr . On integration this gives A^2 = 2p[-e^(-1/2(r^2))] > evaluated from zero to infinity. Substituting in for the limits - gives > when inifinity (or try a large number) is substituted for r, the > expression yields zero, when zero is substituted for r, then the expression > yields minus 1, which is subtracted from the upper limit, so it becomes > plus 1, giving A^2 = 2p. We wanted to find A, so we take the square root > of this giving, A = sqrt(2p). > > Sorry I didn't have the symbolic representation, but if you write it out it > you should be able to folllow the above line of reasoning. > > In some versions of the above 'trick' you might see that they define the > 'square' base area with co-ordinates x and y, but then replace x & y more > formally with polar co-ordinates r and q giving x = r cos q and y = r sin > q, They then replace the element of area (dx, dy) by (r dr dq), and the > range of integration has to be altered from the square OABC (O = origin, C > = distance along x axis, A = distance along y axis, B = opposite corner to > O) to the quadrant OAC, which results in an error denoted by e. This is the > space between the outer 'box' and the curve joining x (C) and y (A). In > this case the limits of integration are :- outer limit from 0 to R, inner > limit from 0 to p/2. The expression to be evaluated changes to > (e^(- r^2))r dr dq, plus the error term e. The volume represented by e has > a base area which is less than 1/2 R^2, and a maximum height of exp (- > R^2). Thus > > e < 1/2 R^2 exp( - R^2), thus as R goes to infinity, e goes to 0. This > might satisfy Digby? > > Thus A^2 = 1/4p - 1/4 p exp(- R^2) + e > > Therefore, as R goes to infinity, A^2 goes to 1/4p. In this latter case > the algebra had been simplified at the start by removing the normalising > factor, 2/(sqrt(p)), which is used to make erf ( inifinity ) = 1. > > I hope this helps. > > John Butler > > Gerald van den Boogaart <[EMAIL PROTECTED]> on 20/12/2005 09:54:17 > > Please respond to [EMAIL PROTECTED] > > To: "Digby Millikan" <[EMAIL PROTECTED]> > cc: "AI Geostats mailing list" <[email protected]> > > Subject: Re: [ai-geostats] Cumulative gaussian distribution > > 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 > -------------------------------------------------- > > * By using the ai-geostats mailing list you agree to follow its rules > ( see http://www.ai-geostats.org/help_ai-geostats.htm ) > > * To unsubscribe to ai-geostats, send the following in the subject or in > the body (plain text format) of an email message to [EMAIL PROTECTED] > > Signoff ai-geostats > > ********************************************************************** > This transmission is confidential and must not be used or disclosed by > anyone other than the intended recipient. Neither Corus Group Plc nor > any of its subsidiaries can accept any responsibility for any use or > misuse of the transmission by anyone. > ********************************************************************** > > > -------------------------------------------------------------------------------- > * By using the ai-geostats mailing list you agree to follow its rules > ( see http://www.ai-geostats.org/help_ai-geostats.htm ) > > * To unsubscribe to ai-geostats, send the following in the subject or in the > body (plain text format) of an email message to [EMAIL PROTECTED] > > Signoff ai-geostats -- ~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~= From: Prof. Dr. Rudolf Dutter Dept. of Statistics and Probability Theory Vienna University of Technology, Wiedner Hauptstr. 8-10 A-1040 Vienna, Austria Tel. +43 1 58801/10730 FAX +43 1 58801/10799 E-Mail: [EMAIL PROTECTED] Internet: http://www.statistik.tuwien.ac.at/public/dutt/ ~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=
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