The Normal Distribution: A derivation from basic principles


 Wilfred Heath
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1 Th Normal Distribution: A drivation from basic principls Introduction Dan Tagu Th North Carolina School of Scinc and Mathmatics Studnts in lmntary calculus, statistics, and finit mathmatics classs oftn larn about th normal curv and how to dtrmin probabilitis of vnts using a tabl for th standard normal probability dnsity function Th calculus studnts can wor dirctly xµ I with th normal probability dnsity function pbxg K J σ and us numrical σ π intgration tchniqus to comput probabilitis without rsorting to th tabls In this articl, w will giv a drivation of th normal probability dnsity function suitabl for studnts in calculus Th broad applicability of th normal distribution can b sn from th vry mild assumptions mad in th drivation Basic Assumptions Considr throwing a dart at th origin of th Cartsian plan You ar aiming at th origin, but random rrors in your throw will produc varying rsults W assum that: th rrors do not dpnd on th orintation of th coordinat systm rrors in prpndicular dirctions ar indpndnt This mans that bing too high dosn't altr th probability of bing off to th right larg rrors ar lss lily than small rrors In igur, blow, w can argu that, according to ths assumptions, your throw is mor lily to land in rgion A than ithr B or C, sinc rgion A is closr to th origin Similarly, rgion B is mor lily that rgion C urthr, you ar mor lily to land in rgion than ithr D or E, sinc has th largr ara and th distancs from th origin ar approximatly th sam igur
2 Dtrmining th Shap of th Distribution Considr th probability of th dart falling in th vrtical strip from x to x + x Lt this probability b dnotd p( x) x Similarly, lt th probability of th dart landing in th horiontal strip from y to y + y b p( y) y W ar intrstd in th charactristics of th function p rom our assumptions, w now that function p is not constant In fact, th function p is th normal probability dnsity function igur rom th indpndnc assumption, th probability of falling in th shadd rgion is p( x) x p( y) y Sinc w assumd that th orintation dosn't mattr, that any rgion r units from th origin with ara x y has th sam probability, w can say that This mans that p( x) x p( y) y g( r) x y g( r) p( x) p( y) Diffrntiating both sids of this quation with rspct to θ, w hav p x dp ( y ) + p y dp ( ( ) x ) ( ) dθ dθ, sinc g is indpndnt of orintation, and thrfor, θ Using x r cosbθg and y r sinbθg, w can rwrit th drivativs abov as c b gh b g p( x) p ( y) r cos θ + p( y) cr sin θ h Rwriting again, w hav p( x) p ( y) x p( y) p ( x) y This diffrntial quation can b solvd by sparating variabls, p ( y) y p( y)
3 This diffrntial quation is tru for any x and y, and x and y ar indpndnt That can only happn if th ratio dfind by th diffrntial quation is a constant, that is, if p ( y) C y p( y) Solving C, w find that Cx p( x) Cx and ln bp( x) g + c and finally, p( x) A C x Sinc w assumd that larg rrors ar lss lily than small rrors, w now that C must b ngativ W can rwrit our probability function as with positiv p( x) A x, This argumnt has givn us th basic form of th normal distribution This is th classic bll curv with maximum valu at x and points of inflction at x now nd to dtrmin th appropriat valus of A and Dtrmining th Cofficint A ± W or p to b a probability distribution, th total ara undr th curv must b W nd to adjust A to insur that th ara rquirmnt is satisfid Th intgral to b valuatd is A x dx x x If A dx, thn dx Du to th symmtry of th function, this ara A is x twic that of dx, so Thn, x dx A I x y KJ H G I K J dx dy, 4 A 3
4 sinc x and y ar just dummy variabls Rcall that x and y ar also indpndnt, so w can rwrit this product as a doubl intgral x + y j dy dx 4 A (Rwriting th product of th two intgrals as th doubl intgral of th product of th intgrands is a stp that nds mor justification than w giv hr, although th rsult is asily blivd It is straightforward to show that I KJ H G I K J M M M M f ( x) dx g( y) dy f ( x) g( y) dy dx for finit limits of intgration, but th infinit limits crat a significant challng that will not b tan up) Th doubl intgral can b valuatd using polar coordinats x + y π / dx dy r j r dr dθ To valuat th polar form rquirs a usubstitution in an impropr intgral Prforming th intgration with rspct to r, w hav O r L u dθ π r dr dθ du d N M P θ π / π / π / Q Now w now that π, and so A Th probability distribution is 4 A π Dtrmining th Valu of p( x) x π A qustion oftn asd about probability distributions is "what ar th man and varianc of th distribution?" Prhaps th valu of has somthing to do with th answr to ths qustions Th man, µ, is dfind to b th valu of th intgral dx Th varianc, σ, is th valu of th intgral bx µ g p( x) dx Sinc th function is an odd function, w now th man is ro Th valu of th varianc nds furthr computation 4
5 To valuat dx σ, w procd as bfor, intgrating on only th positiv xaxis and doubling th valu Substituting what w now of p( x), w hav x x dx σ π Th intgral on th lft is valuatd by parts with u x and dv x xprssion M L x x x M O lim + dx π M P Simplifying, w now that lim M N x x from our wor bfor So x π M x Th Normal Probability Dnsity unction Q x to gnrat th and w now that x π dx π dx so that π σ Now w hav th normal probability distribution drivd from our 3 basic assumptions: x pbxg H G I K J σ σ π Th gnral quation for th normal distribution with man µ and standard dviation σ is cratd by a simpl horiontal shift of this basic distribution, Rfrncs: pbxg σ π xµ σ I K J Grossman, Stanly, I, Multivariabl Calculus, Linar Algbra, and Diffrntial Equations, nd, Acadmic Prss, 986 Hamming, Richard, W Th Art of Probability for Enginrs and Scintists, AddisonWsly, 99 5
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