Removing magic from the normal distribution and the Stirling and Wallis formulas.

Transcription

1 Removig magic from the ormal distributio ad the Stirlig ad Wallis formulas. Mikhail Kovalyov, Published i Mathematical Itelligecer, 0, Volume 33, Number 4. The Wallis formula is ofte writte as. Itroductio = π (.a) with the meaig that the sequece of partial products p =, p = 3, p 3 = 3, p 4 = 5, p 4 = 5 6,... of the left-had 5 side of (.a) coverges to π. Sice the partial product of the first terms ] (!) p = ()! we may rewrite (.a) as + (!) lim ()! = π, (.b) which is ofte used to obtai the ubiquitous Stirlig formula lim m e m m! m m m = π, (.) described i McCarti, 006 as providig a itriguig coectio betwee π ad e. To obtai (.) we rewrite (.b) as (f m ) lim = π, f m = em m! m f m m m m. (.3)

2 ( ) m+0.5 f m+ e m Sice f m > 0, = ( ) f m + m+0.5 = e <, sequece f m is positive ad mootoically decreasig ad as such must have a o-egative m+ m limit. To show that lim f m 0, cosider sequece g m = (m )f m which has m m the same limit as sequece f m. Sice g m+ = e ( ) m+0.5 m m >, the g m m m+ lim g m > g > 0 ad thus lim f m > 0. Formula () is obtaied by sim- m m ple applicatio of the laws of limits to (.3) as follows π = ( ) lim f m m lim m f m (f m ) lim = m f m = lim m f m. Thus (.) ad (.) are equivalet i the sese that each oe of them implies the other. Of may applicatios of the Stirlig formula oe is the derivatio of the ormal distributio as the limitig case of the biomial distributio. I a typical derivatio foud i may textbooks, oe cosiders a ifiite row of cells umbered by itegers ad a oe-dimesioal radom walk of a poit P that starts at the cell K=0 ad at each step jumps from the poit it occupies to either the right or left adjacet cell with probability The probability of fidig poit P at a cell K after N steps is give by the followig table: K= N = N = N = N = N = N = N = N = The ozero etries are multiples of the biomial coefficiets N! ( N N+K! ) ( N K! ), K N writte as eve fuctios of K. The probability P (K, N) of fidig

3 3 poit P at a cell labeled K is give by N! P (K, N) = ( N N+K! ) ( ), if K N; N K is eve, N K! 0, otherwise. (.4) For N sufficietly large ad K N, formula () allows us to approximate N! πnn N e N N ± K,! ( ) N±K N ± K π(n ± K) e N±K which N! upo substitutio ito ( N N+K! ) ( N K! ) lead to approximatio N! ( N N+K! ) ( N K! ) N of the biomial distributio by the ormal distributio. πn e K The give derivatio of the ormal distributio is based o formula (.) with all difficulties buried i (.), or equivaletly i (.). Oe would assume that formulas as fudametal as (.) ad (.) had a ituitive proof, yet as poited out i Gowers (008), all proofs of (.) seem to cotai a o-ituitive step with a idetity or a estimate magically pulled out of a hat. Attempts to fid a simple ituitive proof have led to a rather large umber of publicatios some of which are listed i Bibliography, yet oe seems to be fully ituitive. Most assume that formulas (.) are kow ad try to costruct a appropriate proof. But is it possible to arrive at formulas (.) i a completely atural way without ay magical steps? The author of this paper thiks it is ad will show how i the ext sectio based o the ideas outlied i Kovalyov (009).. Ituitive derivatio of the Wallis formula ad the ormal distributio. For simplicity s sake let us take The P ( k, )=P (k, )= N = is eve. (.) ()! (!) (!) ( + k)!( k)! = ()! (!) k j= (k j) + j = ()! k k j (!) + j, if 0 k, j= 0, if k >. (.)

4 4 Coefficiets P (k, ) also satisfy P (k, ) = k= k= ( ()! ( + k)!( k)! = + ) =. (.3) k= = due to (.3) k= approaches π as The mai idea of the derivatio of (.b) is to estimate P (k, ) i (.) by usig rather ituitive approximatios k j e j k, + j e k j j, + j e k k k j which lead to j= + j e (!) k, P (k, ) e k ad ()! (!) ()! P (k, ) e k which, i tur, imply (.). The prob- lem is that the approximatios are valid oly for j k, k j k ad as k gets close to ± the approximatios lose their validity for values of j ad k j close to. To tur these ideas ito a rigorous proof we break up the set k ito the core k ε+0.5 ad two tails ε+0.5 < k, with 0 < ε < 0.5 to be determied later, so that iside the core approximatios k j e j k, + j e j are valid while the total probability of P beig outside the core approaches zero as ε ε }{{}}{{}}{{} left tail the core cosists of itegers k such that k 0.5+ε right tail k< 0.5+ε 0.5+ε <k For large the total probability of P beig i oe of the two tails is ( ) ( ) ( ) P k, = P k, = P k, ε, (.4) k < 0.5+ε k 0.5+ε k 0.5+ε ad hece goes to 0 as. Ideed, takig for simplicity s sake k > 0, P ( k, ) = ()! k (k j) (!) < 0.5 k k + j = + j + j k 0.5+ε }{{} 0.5+ε <k j= 0.5+ε <k j= this term is less tha k ( k ) 0.5 k ( k ) = 0.5 ( + k ) ] k + j + k 0.5+ε <k j= }{{} 0.5+ε <k j= 0.5+ε <k }{{} replacig j with k replacig k with makes it 0.5+ε makes it larger larger

5 ε <k ( ε ) 0.5 ε ] ε 0.5 ( + Iside the core P (k, ) satisfies 0.5 ε ) 0.5 ε e < for large ] ε 0.5+ε <k < ε. e k e 0.5+3ε (!) P (k, ) e k e 0.5+3ε, ()! if k ε+0.5. (.5) Due to P ( k, ) = P (k, ) it suffices to prove (.5) for k 0. To do so we employ iequality e x x + x e x, for x. (.6a) To prove (.6a) otice that fuctios w (x) = e x x, w (x) = + x e x x are aalytic ad satisfy w (0) = w (0) = w (0) = w (0) = 0, w (0) = w (0) =. Thus each of them must of the form 0.5x + o(x ) > 0 for x sufficietly small. Iequality (.6a) applied with x = k j ad x = j gives us e k j k j ( ) k j e k j, (.6b) e j ( j ) + j e j. (.6c) Dividig (.6b) by (.6c) we obtai e k (k j) k j + j e k + j, (.6d) which yields e k k (k j) j= =e k ] k + (k j) j= k j= k j + j e k j= ] k + j =e k + k j j=. (.6e)

6 Usig k (k j) = k(k )(k ) = k3 3k +k 6 3 j= k j k(k + )(k + ) = = k3 + 3k + k 6 6 j= 6 k3 6 (0.5+ε ) ε, k3 (0.5+ε ) ε valid for 0<k 0.5+ε we may further simplify (.6e) to (.5). Multiplyig (.5) by ad summig up i k we obtai 0.5 e k k 0.5+ε which upo divisio by If we take the 0.5 ]e 0.5+3ε (!) e k k 0.5+ε ()! k 0.5+ε P (k, ) becomes k 0.5+ε ] e 0.5+3ε P (k, ) k 0.5+ε lim e 0.5+3ε (.3) ad (.4), lim (!) ()! = lim e 0.5+3ε k 0.5+ε, k is eve P (k, ) e k k 0.5+ε e k k 0.5+ε ] e 0.5+3ε P (k, ) k 0.5+ε ] e 0.5+3ε (.7a). (.7b) 0 < ε < (.8) 6 =, lim P (k, ) = due to k 0.5+ε e k = e t dt = π ad as the limits of the first ad third terms i (.7b) exist ad are equal to π. By the well-kow theorem of Calculus the limit of the middle term (!) ()! must also exist ad be equal to π thus provig (.b). Notice that the idetity e t dt = π comes from multivariable calculus, its proof follows from the strig of idetities e t dt = e x dx e y dy = x e y dxdy = π e r rdrdφ = π e r rdr dφ = π. R = 0 }{{} =π

7 7 We may rewrite (.5) as e ()! π 0.5+3ε (!) P (k, ) π e k e 0.5+3ε ()! π (!). (.9) Sice the first ad third terms of (.9) approach as, we coclude that lim P (k, ) π e k =, (.0a) ad cosequetly P (k, ) π e k, providig us with the simplest case of the Cetral Limit Theorem. Ackowledgemets. (.0b) The author would like to thak the reviewer ad Dr. Chadler Davis for suggestio o improvig the mauscript. Bibliography Brothers, H., Kox, J, 998. New closed-form approximatio to the logarithmic costat e, The Mathematical Itelligecer, Vol. 0, pp Bru, V., 95. Walliss og Brouckers formler for π (i Norwegia), Norsk matematisk tidskrift, Vol. 33, pp Diacois, P., Freedma, D., 986. A Elemetary Proof of Stirlig s Formula, The America Mathematical Mothly, Vol. 93, No., pp Colema, A. J., 95. A Simple Proof of Stirlig s Formula,The America Mathematical Mothly, Vol. 58, No. 5, pp Feller, W., 968. Stirlig s Formula,.9 i A Itroductio to Probability Theory ad Its Applicatios, Vol., 3rd ed. New York: Wiley, pp Feller, W., 967. A Direct Proof of Stirlig s Formula, The America Mathematical Mothly, Vol. 74, No. 0, pp Gowers, T., Kovalyov, M., 009. Elemetary Combiatorial-Probabilistic Proof of the Wallis ad Stirlig Formulas, Joural of Mathematics ad Statistics, Vol. 5, Issue 4, pp

8 McCarti, B., 006, e: The Master of All, The Mathematical Itelligecer, Vol. 8, No., pp. 0-. Mavromatis, H., 99. The Wallis formula for π, icludig two ew expressios ivolvig irratioals, Iteratioal Joural of Computer Mathematics, Vol. 43, Issue 3 ad 4, pp Miller, S., 008. A probabilistic proof of Wallis s formula for π, The America Mathematical Mothly, Vol. 5, No. 8, pp Moritz, R. E., 98. A Elemetary Proof of Stirlig s Formula,Joural of the America Statistical Associatio, Vol. 3, No. 6, pp Mortici, 00. New improvemets of the Stirlig formula, Applied Mathematics ad Computatio, Vol. 7, Issue, 5, pp Robbis, H., 955. A Remark of Stirlig s Formula, The America Mathematical Mothly, Vol. 6, 6-9. Romik, D., 000. Stirlig s approximatio for!: the ultimate short proof?,the America Mathematical Mothly, Vol. 07, pp Shozo Niizeki; Makoto Araki, 00. Simple ad clear proofs of Stirlig s formula, Iteratioal Joural of Mathematical Educatio i Sciece ad Techology, Vol. 4, Issue 4, pp Sodow, J., 005. A faster product for π ad a ew itegral for l π America Mathematical Mothly, Volume, pp , The Wastlud, J., 005. A elemetary proof of Wallis product formula for π, Likopig studies i mathematics, No.. Yaglom, A. M. ad Yaglom, I. M., 953. A elemetary derivatio of the formulas of Wallis, Leibitz ad Euler for the umber π (i Russia), Uspechi matematiceskich auk, 8, o. 5, 57, pp