Topics in Probability Theory and Stochastic Processes Steven R. Dunbar. Evaluation of the Gaussian Density Integral

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1 Steve R. Dubar Departmet of Mathematics 3 Avery Hall Uiversity of Nebraska-Licol Licol, NE Voice: Fax: Topics i Probability Theory ad Stochastic Processes Steve R. Dubar Evaluatio of the Gaussia Desity Itegral Ratig Mathematically Mature: proofs. may cotai mathematics beyod calculus with 1

2 Sectio Starter Questio Is it possible to evaluate 1 π e x dx directly by stadard itegratio techiques? Key Cocepts 1. 1 π e x dx = 1.. Several proofs use double itegratio i clever ways. 3. Aother proof uses iterpolatio betwee two stadard itegrals. 4. Yet aother proof uses a approximatio which ca be evaluated usig stadard itegratio techiques. The approximate evaluatios lead to a sequece with a limit related to Wallis Formula. Vocabulary 1. The stadard Gaussia desity fuctio is 1 π e x, < x <.

3 Mathematical Ideas The Gaussia desity itegral The stadard Gaussia desity fuctio is 1 e x, < x <. π I order to claim this is truly a probability desity fuctio, we must show that the total area uder this fuctio is 1. Theorem 1. 1 π e x dx = 1. We provide several proofs of this evaluatio. Accordig to Peter Lee, [], the first complete evaluatio of this itegral was by P. S. Laplace i 1774 startig from a more complicated itegral, usig some chages of variables ad limits, ad fially reachig a special case which ca be reduced to the Gaussia itegral. Lee [] gives 8 methods of evaluatio, icludig 3 cited here alog with other methods usig cotour itegratio ad the Gamma Fuctio. Evaluatio with double itegratio The first proof is the usual proof, relyig o itegratio i polar coordiates. Proof. Set J = e x dx = J = = y e π = π. dy. The we fid that e (x +y dx dy e r r dr dθ 3

4 The J = π. Accordig to Peter Lee, [], this method is due to S. D. Poisso ad was popularized by J. K. F. Sturm. A secod proof usig double itegrals is also due to P. S. Laplace, []. Proof. Note that K = e z / dz = e (xy / y dx for ay y by settig z = xy. Puttig z i place of y, it follows that for ay z so that ( K = = K = e (zx / z dx ( e z / dz e (x +1z / z dz dx. e (zx / z dx Now set (1 + x z = t so that z dz = dt /(1 + x to get K = e t dt (1 + x dx ( ( = e t dx dt (1 + x = ( e t (arcta x = π ad hece K = π/. Fially double the itegral usig symmetry e y / dy = π. A third double-itegratio method due to S. P. Eveso i 5 is cited by P. S. Lee []. 4

5 Proof. Cosider the volume uder the surface z = e (x +y, which is give by V = e (x +y dx dy ( = e dx x. However V ca also be thought of as a volume of revolutio created by revolvig e x about the z axis. Use the disk method of evaluatio for a volume of revolutio. Sice z = e x we have x(z = log z is the radius of a disk at level z. ad hece V = π x dz = π Chagig variables x = y/, log z dz = π[z z log z] 1 = π. e x dx = π. e y / dy = π. Evaluatio by iterpolatio betwee two itegrals Aother proof uses a iterpolatio betwee two itegrals. Proof. Defie ( t H(t = e dx x + for t >. Compute the derivative ( t H (t = e x dx e t + ( t = e x dx e t + e t (x +1 x + 1 e t (x +1 x + 1 dx ( te t (x +1 dx t(x + 1 dx 5

6 Chage variables i the first itegral x = ty, so the ew limits of itegratio are ad 1 ad dx = t dy. After this chage of variables ( H (t = t e t y dy e t + ( te t (x +1 dx ( = t e t x dx e t te t e t x dx =. Evaluatig H at, H( = 1 x + 1 dx = π 4. As t, ( lim H(t = e dx x. t Puttig these all together, e x dx = π. Chagig variables agai, x = y/, e y / dy = Fially double the itegral usig symmetry π. e y / dy = π. Evaluatio usig Wallis Formula Lemma. For > 1 ad all x [, ], e x ( 1 x e 1. 6

7 Proof. To prove the left iequality is equivalet to showig ( 1 x ( e x/ for x [, ]. It suffices to show ( 1 x e x/ or (1 t e t for t [, 1]. This follows immediately sice 1 t is the expressio for the taget lie to e t at the poit (, 1 ad e t is cocave up. To prove the right iequality, let F (x = e x (1 x. Note that F ( =, F ( = e, ad F ( = e. The fuctio F attais a maximum at some poit x of the iterval [, ]. The maximum is positive, sice F ( > F ( =. The maximum does ot occur at sice F ( <. At the maximum ( e x = 1 x 1. The usig this we ca rewrite ( F (x = e x e x 1 x = x e x The fuctio xe x has a maximum value of e 1 at x = 1, so F (x F (x e 1. Now we use this lemma to approximate the Gaussia desity itegral. Proof. Approximate the itegrad e x i the Gaussia desity itegral with (1 x / o the iterval [, ]. [ ] e x (1 x e 1 dx dx. Rewrite this as e x dx 7 (1 x dx e 1.

8 Therefore, takig the limit as, e x dx = lim (1 x dx. Now evaluate the itegral o the right side usig the substitutio x = si t (1 x dx = π/ cos +1 t dt. Cosider J k = π/ cos k (x dx. The itegratig by parts just as i Wallis Formula we get kj k = (k 1J k. Now J 1 = 1 so recursively J 3 =, 3 J 5 = 4 ad iductively 3 5 Hece, e x J +1 = Now use Wallis Formula lim 4 ( ( 1 3 ( 1 ( ( ( dx = lim 1 3 ( 1 ( + 1. ( ( ( ( 1 ( + 1 = π. The e x = lim dx = 1 π ( ( ( 1 ( + 1 Fially chagig variables ad usig symmetry as before 1 π e y dy = 1. 8

9 Sources The first double-itegral proof is stadard ad occurs i may sources. The iterpolatio proof is from a footote i the article by Michel, [4], where it is credited to exercises i Apostol, [1]. The proof usig Wallis Formula is from the short article by Levrie ad Daems, [3]. Problems to Work for Uderstadig Readig Suggestio: Refereces [1] Tom M. Apostol. Mathematical Aalysis. Addiso-Wesley, secod editio, [] Peter M. Lee. The probability itegral. maths/histstat/ormal_history.pdf. [Olie; accessed 15-October- 11]. 9

10 [3] Paul Levrie ad Walter Daems. Evaluatig the probability itegral usig Wallis product formula for π. America Mathematical Mothly, 116(6: , Jue-July 9. [4] Reihard Michel. O Stirlig s formula. America Mathematical Mothly, 19(4:388 39, April. Outside Readigs ad Liks: I check all the iformatio o each page for correctess ad typographical errors. Nevertheless, some errors may occur ad I would be grateful if you would alert me to such errors. I make every reasoable effort to preset curret ad accurate iformatio for public use, however I do ot guaratee the accuracy or timeliess of iformatio o this website. Your use of the iformatio from this website is strictly volutary ad at your risk. I have checked the liks to exteral sites for usefuless. Liks to exteral websites are provided as a coveiece. I do ot edorse, cotrol, moitor, or guaratee the iformatio cotaied i ay exteral website. I do t guaratee that the liks are active at all times. Use the liks here with the same cautio as you would all iformatio o the Iteret. This website reflects the thoughts, iterests ad opiios of its author. They do ot explicitly represet official positios or policies of my employer. Iformatio o this website is subject to chage without otice. Steve Dubar s Home Page, to Steve Dubar, sdubar1 at ul dot edu Last modified: Processed from L A TEX source o October, 11 1