Lecture 3 Gaussian Probability Distribution


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1 Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike the binomil nd Poisson distribution, the Gussin is continuous distribution: P(y) (ym) s p e s m men of distribution (lso t the sme plce s mode nd medin) s vrince of distribution y is continuous vrible ( y ) l Probbility (P) of y being in the rnge [, b] is given by n integrl: P( < y ) s p e b (ym) s Krl Friedrich Guss u The integrl for rbitrry nd b cnnot be evluted nlyticlly + The vlue of the integrl hs to be looked up in tble (e.g. Appendixes A nd B of Tylor). P(x) p(x) (x m )  s p e s gussin Plot of Gussin pdf x K.K. Gn L3: Gussin Probbility Distribution
2 l The totl re under the curve is normlized to one. + the probbility integrl: P( < y < ) s s p l We often tlk bout mesurement being certin number of stndrd devitions (s) wy from the men (m) of the Gussin. + We cn ssocite probbility for mesurement to be m  ns from the men just by clculting the re outside of this region. ns Prob. of exceeding ±ns e (ym) It is very unlikely (< 0.3%) tht mesurement tken t rndom from Gussin pdf will be more thn ± 3s from the true men of the distribution. Reltionship between Gussin nd Binomil distribution l The Gussin distribution cn be derived from the binomil (or Poisson) ssuming: u p is finite u N is very lrge u we hve continuous vrible rther thn discrete vrible l An exmple illustrting the smll difference between the two distributions under the bove conditions: u Consider tossing coin 0,000 time. p(heds) 0.5 N 0,000 K.K. Gn L3: Gussin Probbility Distribution
3 n For binomil distribution: men number of heds m Np 00 stndrd devition s [Np(  p)] / + The probbility to be within ±s for this binomil distribution is: ! P Â m00 ( m m m)!m! n For Gussin distribution: m+s P(m s < y < m +s ) e (ym) s ª 0.68 s p ms + Both distributions give bout the sme probbility! Centrl Limit Theorem l Gussin distribution is importnt becuse of the Centrl Limit Theorem l A crude sttement of the Centrl Limit Theorem: u Things tht re the result of the ddition of lots of smll effects tend to become Gussin. l A more exct sttement: u Let Y, Y,...Y n be n infinite sequence of independent rndom vribles ech with the sme probbility distribution. u Suppose tht the men (m) nd vrince (s ) of this distribution re both finite. Actully, the Y s cn be from different pdf s! + For ny numbers nd b: lim P < Y +Y +...Y n  nm næ Î Í s n b p e y + C.L.T. tells us tht under wide rnge of circumstnces the probbility distribution tht describes the sum of rndom vribles tends towrds Gussin distribution s the number of terms in the sum Æ. K.K. Gn L3: Gussin Probbility Distribution 3
4 + Alterntively: lim P < Y  m næ s / n Î Í lim P < Y  m Í b næ Î s m p e y n s m is sometimes clled the error in the men (more on tht lter). l For CLT to be vlid: u m nd s of pdf must be finite. u No one term in sum should dominte the sum. l A rndom vrible is not the sme s rndom number. u Devore: Probbility nd Sttistics for Engineering nd the Sciences: + A rndom vrible is ny rule tht ssocites number with ech outcome in S n S is the set of possible outcomes. l Recll if y is described by Gussin pdf with m 0 nd s then the probbility tht < y is given by: b  y P( < y ) e p l The CLT is true even if the Y s re from different pdf s s long s the mens nd vrinces re defined for ech pdf! u See Appendix of Brlow for proof of the Centrl Limit Theorem. K.K. Gn L3: Gussin Probbility Distribution 4
5 l Exmple: A wtch mkes n error of t most ±/ minute per. After one yer, wht s the probbility tht the wtch is ccurte to within ±5 minutes? u Assume tht the dily errors re uniform in [/, /]. n For ech, the verge error is zero nd the stndrd devition / minutes. n The error over the course of yer is just the ddition of the dily error. n Since the dily errors come from uniform distribution with well defined men nd vrince + Centrl Limit Theorem is pplicble: lim P < Y +Y +...Y n  nm næ Î Í s n b p e y + The upper limit corresponds to +5 minutes: b Y +Y +...Y n  nm s n The lower limit corresponds to 5 minutes: Y +Y +...Y n  nm s n The probbility to be within ± 5 minutes: P 4.5 e  y p less thn three in million chnce tht the wtch will be off by more thn 5 minutes in yer! K.K. Gn L3: Gussin Probbility Distribution 5
6 l Exmple: Generte Gussin distribution using rndom numbers. u Rndom number genertor gives numbers distributed uniformly in the intervl [0,] n m / nd s / u Procedure: n Tke numbers (r i ) from your computer s rndom number genertor n Add them together n Subtrct 6 + Get number tht looks s if it is from Gussin pdf! P < Y +Y +...Y n  nm Î Í Í PÍ < Í Î Í Â i s n r i  PÍ 6 < Â r i  6 < 6 Î p i 6 e  y 6 Thus the sum of uniform rndom numbers minus 6 is distributed s if it cme from Gussin pdf with m 0 nd s. A) 00 rndom numbers B) 00 pirs (r + r ) of rndom numbers C) 00 triplets (r + r + r 3 ) of rndom numbers D) 00 plets (r + r + r ) of rndom numbers. E) 00 plets K.K. Gn L3: Gussin Probbility Distribution 6 E (r + r + r  6) of rndom numbers. Gussin m 0 nd s
7 l Exmple: The dily income of "crd shrk" hs uniform distribution in the intervl [$40,$]. Wht is the probbility tht s/he wins more thn $0 in 60 s? u Lets use the CLT to estimte this probbility: lim P < Y +Y +...Y n  nm næ Î Í s n b p e y u The probbility distribution of dily income is uniform, p(y). + need to be normlized in computing the verge dily winning (m) nd its stndrd devition (s). m s yp(y) p(y) [  () ] 5  () y p(y)  m 3 [ 3  () 3 ] () p(y) u The lower limit of the winning is $0: Y +Y +...Y n  nm s n u The upper limit is the mximum tht the shrk could win ($/ for 60 s): b Y +Y +...Y n  nm s n P 3.4 e  y ª p p e y % chnce to win > $0 in 60 s K.K. Gn L3: Gussin Probbility Distribution 7
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