A Note on Approximating. the Normal Distribution Function

Transcription

1 Applid Mathmatical Scincs, Vol, 00, no 9, A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and malodat@yuidujo Abstract In this papr, w propos a on-trm-to-calculat approimation to th normal cumulativ distribution function Our approimation has a maimum absolut rror of W compar our approimation to th act on Kywords: Cumulativ distribution function, normal distribution Introduction Lt X b a standard normal random variabl, i, a random variabl with th following probability dnsity function f ( ), - < < Th cumulativ distribution function of th standard normal is givn by t Φ( ) dt () Th last intgral has no closd form Most basic statistical books giv th valus of this intgral for diffrnt valus of in a tabl calld th standard normal tabl From th this tabl w can also find th valu of whn Φ () is known Svral authors gav approimations for by polynomials (Chokri, 003; Johnson, 994; Baily, 9; Polya, 945) Ths approimations giv quit high accuracy, but computr programs ar ndd to obtain thir valus and thy hav a maimum absolut rror of mor than 003 But only th Polya's approimation

2 46 K M Aludaat and M T Alodat Φ ( ) 05( + ) has on-trm-to-calculat whil th othrs nd mor than on trm Thy ar rviwd in Johnson t al (994) as follows: Φ( ) 05( a + a + a3 + a4 + a5 + a6 ), whr a , a , a , a , a , a ( ) ( ) / Φ p( ), 5 5 p(y) 3 Φ 3( ), y 079( ) + p(y) (3 + 35) Φ4( ) 05p 703/ + 65 All ths approimations ar good but thy nd computr program to b obtaind In addition thir invrs can not b obtaind asily In this short not, w propos nw approimations for Φ () and it's invrs W can also find it's maimum absolut rror Th approimation Sinc Φ () is symmtric about zro, it is sufficint to approimat t 0 dt only for all valus of > 0 According to Johnson t al (994) th Ploya's approimation rprsnts an uppr bound for Φ (), i, Φ( ) 05 + If w can find a sharpr uppr bound on Φ (), thn w can improv th Polya's approimation Sinc implis that, thn So th trm

3 Normal distribution function is closr to Φ () than 05 + So w propos th following approimation for Φ () : Φ 5( ) Th invrs cumulativ distribution function is approimatd by log( ( p 05) ), whr p Φ() Our approimation has a maimum absolut rror of To prov this, w nd to study th diffrnc btwn th Φ () and So w dfin for > 0, a function g () as follows t dt 05 g( ) 0 Th first drivativ of g () is g ( ) W usd th Mathmatica Softwar to find th roots of th first drivativ, i, th roots of g ( ) 0 W got th following thr roots 0, and Tsting th drivativ for th sign lads to th following facts: g () is incrasing in th intrval [0, ] [ , [ and g () is dcrasing in [ , ] So g () has th two absolut trm valus and So th absolut maimum rror is

4 4 K M Aludaat and M T Alodat 3 Comparison In this sction, w compar th act valu of Φ () with its approimatd on For positiv valus of, Figur shows th valus of Φ( ) 5 and its approimation against > 0 W s from th Figur that Φ( ) 5 is vry clos to its approimation, which mans that our approimation is vry accurat Tabl Comparison btwn Φ () and its approimations Φ () Φ ( ) Φ ( ) Φ ( ) Φ ( ) Φ ( ) From Tabl w s that our approimation is good Th bst approimation is Φ 3( ), but computing its invrs rquirs computr programs, whil th invrs of our approimation nds simpl algbraic calculations W not that th approimation Φ ( ) is valid only for Figur Th act valu of Φ() -05 (dottd) and its approimation (smooth)

5 Normal distribution function 49 4 Conclusion In this papr, w proposd an approimation to th cumulativ distribution function of th standard normal distribution Our approimation is on-trm-to calculat and is bttr than th Polya's approimation Numrical comparison shows that our approimation is vry accurat Morovr, it dos not rquir computr programs to calculat both cumulativ distribution function and its invrs Acknowldgmnts This rsarch has bn supportd by a grant from Yarmouk Univrsity" Rfrncs [] Baily, B J R (9) Altrnativs to hasting's approimation to th invrs of th normal cumulativ distribution function Applid statistics, 30, (3) [] Chokri (003) A short not on th numrical approimation of th standard normal cumulativ distribution and its invrs Ral 03-T-7, onlin manuscript [3] Johnson, N I, Kotz, S and Balakrishnan, N (994) Continuous univariat distributions John wily & sons [4] LBouf, C, Gugand, J, Roqu, J L and Landry, P (97) Probabilits Ellipss [5] Polya, G (945) Rmarks on computing th probability intgral in on and two dimnsions Procding of th first Brkly symposium on mathmatical statistics and probability, 63-7 [6] Rnyi, A (970) Probability thory North Holland Rcivd: August, 007