Introduction Lecture 3 Gussin Probbility Distribution Gussin probbility distribution is perhps the most used distribution in ll of science. lso clled bell shped curve or norml distribution Unlike the binomil nd Poisson distribution, the Gussin is continuous distribution: (y$µ) P(y) " # e$ " µ men of distribution (lso t the sme plce s mode nd medin) σ vrince of distribution y is continuous vrible (- y ) Probbility (P) of y being in the rnge [, b] is given by n integrl: P( < y ) " # e$ b (y$µ) " Krl Friedrich Guss 777-855 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 #µ ) #! " e! gussin Plot of Gussin pdf x K.K. Gn L3: Gussin Probbility Distribution
The totl re under the curve is normlized to one. the probbility integrl: P("# < y < #) $ We often tlk bout mesurement being certin number of stndrd devitions (σ) wy from the men (µ) of the Gussin. We cn ssocite probbility for mesurement to be µ - nσ from the men just by clculting the re outside of this region. nσ 0.67 0.5 0.3 0.05 3 0.003 4 0.00006 Prob. of exceeding ±nσ e "(y"µ) $ # "# It is very unlikely (< 0.3) tht mesurement tken t rndom from Gussin pdf will be more thn ± 3σ from the true men of the distribution. 95 of re within σ Only 5 of re outside σ Reltionship between Gussin nd Binomil distribution The Gussin distribution cn be derived from the binomil (or Poisson) ssuming: p is finite N is very lrge we hve continuous vrible rther thn discrete vrible K.K. Gn L3: Gussin Probbility Distribution
An exmple illustrting the smll difference between the two distributions under the bove conditions: Consider tossing coin 0,000 time. p(heds) 0.5 N 0,000 For binomil distribution: men number of heds µ Np 00 stndrd devition σ [Np( - p)] / The probbility to be within ±σ for this binomil distribution is: 00+ 0 4! P # (0 4 0.5 m 0.5 04 "m 0.69 m00" " m)!m! For Gussin distribution: µ+# P(µ "# < y < µ +# ) e "(y"µ) # 0.68 # $ µ"# Both distributions give bout the sme probbility! Centrl Limit Theorem Gussin distribution is very pplicble becuse of the Centrl Limit Theorem A crude sttement of the Centrl Limit Theorem: Things tht re the result of the ddition of lots of smll effects tend to become Gussin. A more exct sttement: Let Y, Y,...Y n be n infinite sequence of independent rndom vribles ech with the sme probbility distribution. Suppose tht the men (µ) nd vrince (σ ) of this distribution re both finite. Actully, the Y s cn be from different pdf s! K.K. Gn L3: Gussin Probbility Distribution 3
For ny numbers nd b: lim P < Y +Y +...Y n $ nµ ) n"# '( n * + ḇ, 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. Alterntively: lim P < Y $ µ n"# / n ) ' ( * + lim P < Y $ µ ) ( + ḇ n"# ' m *, e$ y σ m is sometimes clled the error in the men (more on tht lter). For CLT to be vlid: µ nd σ of pdf must be finite. No one term in sum should dominte the sum. A rndom vrible is not the sme s rndom number. Devore: Probbility nd Sttistics for Engineering nd the Sciences: A rndom vrible is ny rule tht ssocites number with ech outcome in S S is the set of possible outcomes. Recll if y is described by Gussin pdf with µ 0 nd σ then the probbility tht < y is given by: b " y P( < y )! e # 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! See Appendix of Brlow for proof of the Centrl Limit Theorem. K.K. Gn L3: Gussin Probbility Distribution 4
Exmple: Generte Gussin distribution using rndom numbers. Rndom number genertor gives numbers distributed uniformly in the intervl [0,] µ / nd σ / Procedure: Tke numbers (r i ) from your computer s rndom number genertor Add them together Subtrct 6 Get number tht looks s if it is from Gussin pdf! $ ' P < Y +Y +...Y n " nµ # n () $ ' + r i " * P < i * ) ) ) ( $ ' P "6 < + r i " 6 < 6 ( ) i 6 -, e" y "6 Thus the sum of uniform rndom numbers minus 6 is distributed s if it cme from Gussin pdf with µ 0 nd σ. 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 -6 0 +6 K.K. Gn L3: Gussin Probbility Distribution 5 E is close to! (r + r + r - 6) of rndom numbers. Gussin µ 0 nd σ
Exmple: A wtch mkes n error of t most ±/ minute per dy. After one yer, wht s the probbility tht the wtch is ccurte to within ±5 minutes? Assume tht the dily errors re uniform in [-/, /]. For ech dy, the verge error is zero nd the stndrd devition / minutes. The error over the course of yer is just the ddition of the dily error. 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 $ nµ ) n"# '( n * + ḇ, e$ y The upper limit corresponds to +5 minutes: b Y +Y +...Y n " nµ 5 " 365 $ 0 # n 365 4.5 The lower limit corresponds to -5 minutes: Y +Y +...Y n " nµ "5 " 365$ 0 "4.5 # n 365 The probbility to be within ± 5 minutes: 4.5 e # y P $ 0.999997 # 30 #6 " #4.5 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 6
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 dys? Lets use the CLT to estimte this probbility: lim P < Y +Y +...Y n $ nµ ) n"# '( n * + ḇ, e$ y The probbility distribution of dily income is uniform, p(y). need to be normlized in computing the verge dily winning (µ) nd its stndrd devition (σ). µ $ # yp(y) # p(y) [ " () ] 5 " () # y p(y) " µ 3 [ 3 " () 3 ] " 5 675 " () # p(y) The lower limit of the winning is $0: Y +Y +...Y n " nµ 0 " 60$ 5 # n 675 60 00 0 The upper limit is the mximum tht the shrk could win ($/dy for 60 dys): b Y +Y +...Y n " nµ 3000 " 60$ 5 # n 675 60 700 0 3.4 P 3.4 e " y ' ( e" y 0.6 6 chnce to win > $0 in 60 dys K.K. Gn L3: Gussin Probbility Distribution 7