Probability theory STATISTICAL MODELING OF MULTIVARIATE EXTREMES, FMSN15/MASM23 TABLE OF FORMULÆ. Basic probability theory
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- Ξανθίππη Ζάππας
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1 Lud Istitute of Techology Cetre for Mathematical Scieces Mathematical Statistics STATISTICAL MODELING OF MULTIVARIATE EXTREMES, FMSN5/MASM3 Probability theory Basic probability theory TABLE OF FORMULÆ Let S be a sample space, ad let P be a probability o S. The, for all evets A, B, A, A,..., A S, ) Kolmogorov s axioms.) 0 PA).) PS) =.3) PA B) = PA) + PB), if A ad B are disjoit. ) PA B) = PA) + PB) PA B). 3) A ad B are idepedet PA B) = PA) PB). PA B) 4) Coditioal probability: PB A) =. PA) 5) Law of total probability: PB) = PB A i ) PA i ), i= wheever A,..., A are pairwise disjoit ad satisfy 6) Bayes theorem: PA i B) = PB A i)pa i ) PB) wheever A,..., A are pairwise disjoit ad satisfy Oe-dimesioal radom variables = A i = S. i= PB A i )PA i ) j= PB A j) PA j ), A k = S. 7) Distributio fuctio for the radom variable X: F X x) = PX x). 8) Probability-mass fuctio for the discrete radom variable X: p X x) = PX = x). 9) Desity fuctio for the cotiuous radom variable X: f X x) = df Xx) for all x where F X is dx differetiable. 0) If X is discrete, the Pa < X b) = F X b) F X a) = x ]a;b] p X x) 0 k= p X x) If there is o elemet x i ]a; b] such that p X x) 0, the F X b) F X a) = 0. ) If X is cotiuous, the Pa < X b) = F X b) F X a) = b a f X x) dx
2 ii TABLE OF FORMULÆ, FMSN5/MASM3, Two-dimesioal radom variables ) Joit distributio fuctio for the two radom variables X ad Y : F X;Y x; y) = PX x Y y) 3) Joit probability-mass fuctio for the two discrete radom variables X ad Y : p X;Y x; y) = PX = x Y = y) 4) Joit desity fuctio for the two cotiuous radom variables X ad Y : f X;Y x; y) = F X;Y x; y) x y for all x; y) where the derivative exists. 5) If X ad Y both are discrete: P X; Y ) A ) = x;y) A p X;Y x;y) 0 p X;Y x; y) If there is o pair x; y) i A such that p X x) 0, the P X; Y ) A ) = 0. 6) If X ad Y both are cotiuous: P X; Y ) A ) = Coditioal distributios x;y) A f X;Y x; y) dx; y) 7) Coditioal distributio fuctio: F X Y x; y) = PX x Y = y). 8) Coditioal probability-mass fuctio for the discrete radom variable X: p X Y x; y) = PX = x Y = y) If Y is also discrete, the p X Y x; y) = p = X;Y x;y) p Y y), p Y y) 0 = 0, p Y y) = 0. 9) Coditioal desity fuctio for the cotiuous radom variable X: f X Y x; y) = F X Y x; y) x If Y is also cotiuous, the f X Y x; y) = f = X;Y x;y) f Y y), f Y y) 0 = 0, f Y y) = 0. 0) Bayes theorem: = p X Y x;y) p Y y) 0.) X discrete ad Y discrete: p Y X y; x) = p X x), p X x) 0, = 0, p X x) = 0. = p X Y x;y) f Y y) 0.) X discrete ad Y cotiuous: f Y X y; x) = p X x), p X x) 0, = 0, p X x) = 0.
3 TABLE OF FORMULÆ, FMSN5/MASM3, iii = f X Y x;y) p Y y) 0.3) X cotiuous ad Y discrete: p Y X y; x) = f X x), f X x) 0, = 0, f X x) = 0. = f X Y x;y) f Y y) 0.4) X cotiuous ad Y cotiuous: f Y X y; x) = f X x), f X x) 0, = 0, f X x) = 0. ) Margial probability-mass fuctio for the discrete radom variable X:.) Y is discrete: p X x) = y.) Y is cotiuous: p X x) = p Y y) 0 p X Y x; y) p Y y) = y p X;Y x;y) 0 p X Y x; y) f Y y) dy ) Margial desity fuctio for the cotiuous radom variable X: p X;Y x; y).) Y is discrete: f X x) = y f X Y x; y) p Y y).) Y is cotiuous: f X x) = p Y y) 0 3) If X ad Y are idepedet, 3.) the F X;Y x; y) = F X x) F Y y), f X Y x; y) f Y y) dy = 3.) the p X;Y x; y) = p X x) p Y y) if X ad Y are discrete, 3.3) the f X;Y x; y) = f X x) f Y y) if X ad Y are cotiuous, 3.4) the F X Y x; y) = F X x), 3.5) the p X Y x; y) = p X x) if X is discrete, 3.6) the f X Y x; y) = f X x) if X is cotiuous. Law of total probability agai Law of total probability: Let A be a evet. 4) If X is a discrete radom variable, the PA) = 5) If X is a cotiuous radom variable, the PA) = Expectatio, variace, ad the like p X x) 0 f X;Y x; y) dy PA X = x) p X x). PA X = x) f X x) dx. 6) Let g be a real-valued fuctio x gx). The the expectatio of gx) is give by 6.) E gx) ) = gx) p X x), if X is discrete, 6.) E gx) ) = p X x) 0 gx) f X x) dx, if X is cotiuous. 7) Let g be a real-valued fuctio x; y) gx; y). The the expectatio of gx; Y ) is give by 7.) E gx; Y ) ) = gx; y) p X;Y x; y), if X ad Y are discrete, p X;Y x;y) 0
4 iv TABLE OF FORMULÆ, FMSN5/MASM3, 7.) E gx; Y ) ) = gx; y) f X;Y x; y) dx; y), X ) ) 8) Variace: VX) = E EX) = EX ) EX) ). if X ad Y are cotiuous. 9) Stadard deviatio: DX) = VX). 30) Coefficiet of variatio: RX) = DX)/EX). 3) Covariace: CX; Y ) = E X EX) ) Y EY ) ) ) = EXY ) EX) EY ). 3) CX; X) = VX). 33) Coefficiet of correlatio: rx; Y ) = CX; Y ) DX) DY ). 34) Expectatio is liear, i.e. EaX + by ) = aex) + bey ). 35) VaX ± by ) = a VX) + b VY ) ± ab CX, Y ). 36) Covariace is biliear, i.e. CaX + by ; cz) = accx; Z) + bccy ; Z) ad CcZ; ax + by ) = cacz; X) + cbcz; Y ). 37) For idepedet radom variables X, Y : EXY ) = EX) EY ). 38) Gauss approximatios: Let g be a real-valued fuctio x ; x ;... ; x ) gx ; x ;... ; x ). The EgX ;... ; X )) gex );... ; EX )). VgX ;... ; X )) c i VX i ) + c i c j CX i ; X j ), i= i j i<j where c i = g x i EX );... ; EX )) = g x i EX );... ; EX )). Normal Gaussia) distributio 39) Uivariate ormal Gaussia) distributio σ > 0): X Nm; σ ) X m σ N0; ) 40) Bivariate ormal Gaussia) distributio: Let m, m, σ, σ, ad ϱ be real umbers σ > 0, σ > 0, < ϱ < ). If X; Y ) Nm ; m ; σ ; σ ; ϱ), the 40.) f X;Y x; y) = pσ σ ϱ e «ρ) x m ) σ + y m ) σ ϱ x m y m σ σ 40.) X Nm ; σ ), Y Nm ; σ ), CX; Y ) = ϱσ σ, rx; Y ) = ϱ, ) ) 40.3) f X Y x; y) = pσ ϱ e σ ϱ x m +ϱ σ y m ) σ ), i.e. a N m + ϱ σ σ y m ); σ ϱ ) ) distributio, 40.4) ax + by Nam + bm ; a σ + b σ + abϱσ σ ) for all real umbers a ad b.,
5 TABLE OF FORMULÆ, FMSN5/MASM3, v Limit theorems 4) Law of Large Numbers LLN): Let X, X,... be idepedet ad idetically distributed radom variables whith existig expectatio EX i ) = m. The då. Y = X X EX i ), 4) Cetral Limit Theorem: Let X, X,..., X be idepedet ad idetically distributed radom variables whith existig expectatio EX i ) ad existig stadard deviatio DX i ) = σ <. The Y = X X AsN m; σ ), whe. 43) We have approximately 43.) Bi; p) Pop) if p 0 ad ) Bi; p) Np; p p)) if p p) ) Pom) Nm; m) if m 5. Sums of radom variables 44) Let X Nm ; σ ),..., X Nm ; σ ) be idepedet, ormally distributed radom variables. For ay set c,..., c of real umbers, we have ) c i X i N c i m i ; c i σ i. i= i= i= 45) If X ad X are idepedet, the Statistics Poit estimatio 45.) X Bi ; p), X Bi ; p) X + X Bi + ; p). 45.) X Pom ), X Pom ) X + X Pom + m ). 45.3) X Gammaa ; b), X Gammaa ; b) X + X Gammaa + a ; b). 45.4) X χ f ), X χ f ) X + X χ f + f ). Let x,...,x be observatios of idepedet, idetically distributed radom variables with expectatio m ad stadard deviatio σ. The ubiased estimatios of m ad σ are give by 46) m = x i = x 47) σ ) = i= x i m), m kow. i= 48) σ ) = s = x i x), m ukow. i=
6 vi TABLE OF FORMULÆ, FMSN5/MASM3, Cofidece itervals 49) Let ϑ be some parameter, ad let J r.v.) be a estimator of ϑ such that J is approximately) ormally distributed with expectatio ϑ. Let ϑ be the estimate of ϑ, i.e. let ϑ be the observatio of J. The I ϑ = [ϑ λ γ)/ dj ); ϑ + λ γ)/ dj )] I ϑ = [ϑ λ γ dj ); ] I ϑ = [; ϑ + λ γ dj )] two-sided), oe-sided, bouded below), oe-sided, bouded above) are cofidece itervals for ϑ with approximative cofidece level γ γ is typically large, γ = 0,95, γ = 0,99,... ). Here, dj ) is the stadard error of the estimator J. dj = DJ ) if DJ ) is kow ad idepedet of ϑ, ) = = DJ ) ) if DJ ) is ukow or depedet o ϑ. Examples ϑ is the parameter): 49.) Observatios: x,..., x. Radom variables lyig behid the observatios: X,..., X. Model: X,..., X are IID with EX ) =... = EX ) = ϑ ad DX ) =... = DX ) = σ. σ is kow ad idepedet of ϑ. The umber is large. Estimate, stadard error: ϑ = x, dj ) = σ, where x = i= x i. 49.) Observatios: x,..., x. Radom variables lyig behid the observatios: X,..., X. Model: X,..., X are IID with EX ) =... = EX ) = ϑ ad DX ) =... = DX ) = σ. σ is ukow but idepedet of ϑ. The umber is large. Estimate, stadard error: ϑ = x, dj ) = s, where x = i= x i, s = i= x i x). 49.3) Observatios: x,..., x. Radom variables lyig behid the observatios: X,..., X. Model: X Nϑ; σ ),..., X Nϑ; σ ) are IID. The stadard deviatio σ is kow ad idepedet of ϑ. Of course, σ > 0. Estimate, stadard error: ϑ = x, dj ) = σ, where x = i= x i. Commet: The itervals will be exact. 49.4) Observatios: x,..., x. Radom variables lyig behid the observatios: X,..., X. Model: X Nϑ; σ ),..., X Nϑ; σ ) are IID. The stadard deviatio σ is ukow but idepedet of ϑ. Of course, σ > 0. Estimate, stadard error: ϑ = x, dj ) = s, where x = i= x i, s = i= x i x). Commet: The itervals will be exact if the stadard-ormal quatile is replaced by a t ) quatile. Otherwise, the approximative iterval is iaccurate uless is large.
7 TABLE OF FORMULÆ, FMSN5/MASM3, vii 49.5) Observatios: x,..., x. Radom variables lyig behid the observatios: X,..., X. Model: X Poϑ),..., X Poϑ) are IID. It is so that ϑ 5. x Estimate, stadard error: ϑ = x, dj ) =, where x = i= x i. Commet: It must be verified that ϑ 5; the use that i= x i = ϑ. 49.6) Observatio: x. Radom variable lyig behid the observatio: X. Model: X Bi; ϑ). It is so that ϑ ϑ) 0. Estimate, stadard error: ϑ = x, ϑ ϑ dj ) = ). Commet: It must be verified that ϑ ϑ ) 0; the use that x x)/ = ϑ ϑ ). Hypothesis testig 50) Let ϑ be a parameter. We wat to test the simple hypothesis H 0 : ϑ = ϑ 0 o the sigificace level α α is typically small, α = 0,05, α = 0,0,... ). The the test will be Reject H 0 ϑ 0 I Do ot reject H 0 ϑ 0 I where I is a cofidece iterval of ϑ with approximative) cofidece level α. If H : ϑ ϑ 0, the choose I to be two-sided. If H : ϑ > ϑ 0, the choose I to be oe-sided ad bouded below. If H : ϑ < ϑ 0, the choose I to be oe-sided ad bouded above. Miscellaeous The Poisso process Let Nt) be the umber of evets takig place i the time iterval ]0; t]. If Nt) is a Poisso process with costat itesity λ, the 5) Nt) Poλt). 5) Time lags betwee cosecutive evets are idepedet ad expoetially distributed with expectatio /λ. 53) The umber of evets occurrig i a time iterval I ad the umber of evets occurrig i aother time iterval I are idepedet if I ad I are disjoit. Quatiles Quatiles is the same as fractiles. Let α be a real umber such that 0 < α <. Let X be a cotiuous radom variable with distributio fuctio F X. 54) The α-quatile deoted x α ) is defied to be ay umber such that PX > x α ) = α or, equivaletly, F X x α ) = α.
8 viii TABLE OF FORMULÆ, FMSN5/MASM3, 55) x /4 = x 0,5 is called the upper distributio) quartile. x / = x 0,5 is called the distributio) media. x 3/4 = x 0,75 is called the lower distributio) quartile. x 0,0, x 0,0,..., x 0,98, x 0,99 are called the distributio) percetiles. 56) Quatiles λ α based o the stadard-ormal distributio are deoted λ α : If X N0; ), the PX > λ α ) = α Φλ α ) = α λ α = Φ α), where Φ...) is the iverse fuctio of Φ. So, Φ...) has othig to do with 56.) Examples: α = 0, λ α =, α = 0,05 λ α =, α = 0,05 λ α =, α = 0,0 λ α =, ) λ α = λ α for all α such that 0 < α < ) Log-ormal distributio Φ...).) 57) Let X be a log-ormally distributed radom variable, i.e. l X Nm; σ ) or l X x / ) N0; σ ). The the coefficiet of variatio is give by DX) EX) = e σ 58) Let X ad X be two idepedet radom variables. The l X Nm ; σ ), l X Nm ; σ ) lx k X k ) Nk m +k m ; k σ +k σ ) Maximum ad miimum Let X,..., X be idepedet, idetically distributed radom variables with distributio fuctio F X x). If we defie the X max = maxx ;... ; X ) ad X mi = mix ;... ; X ), 59) F Xmax z) = F X z) ), 60) F Xmi z) = F X z) ).
9 TABLE OF FORMULÆ, FMSN5/MASM3, ix Distributio Table of distributios 6) Hypergeometric distributio 8 < = N x ) N x) px) = N +N ) ; x Z [ mi;n );mi;n )] : = 0; otherwise Parameter restrictios N Z N Z Z [;N +N ] Expectatio N N +N Variace N +N N +N N N N +N ) 6) Biomial distributio, Bi; p) px) = = ` x px p) x ; x 0; ;... ; } = 0; otherwise Z 0 < p < p p p) 63) Poisso distributio, Pom) px) = m mx = e x! x Z 0 = 0, otherwise m > 0 m m 64) Geometric distributio, Gep) px) = = p p) x ; x Z 0 = 0; otherwise 0 < p < p p p p 65) First success distributio 66) Uiform distributio, Ua; b) px) = = p p) x ; x Z = 0; otherwise = b a fx) = ; a < x < b = 0; otherwise 8 >< = 0; x a F x) = = x a b a >: ; a < x < b = ; x b 0 < p < a < b p a + b p p a b) 67) Beta distributio, ba; b) = Γa+b) Γa) Γb) fx) = xa x) b ; 0 < x < = 0; otherwise 8 >< F x) = = Γa+b) Bx; a; b); 0 < x < Γa) Γb) >: = ; x a > 0, b > 0 a a + b ab a + b) a + b + ) 68) Normal Gaussia) distributio, Nm; σ ) x m) fx) = e σ pσ F x) = Φ ` x m σ σ > 0 m σ 69) Log-ormal distributio, l X Nm;σ ) F x) = l x m = Φ ); x > 0 σ σ > 0 e m+σ / e m+σ e m+σ 70) Log-ormal = 0; distributio, 3, F x) = x 0 = Φ l X N0;σ ) l x ; x > 0 σ x / x / x / >0, σ >0 x / e σ / x / e σ e σ ) Γ is the gamma fuctio; cf. 84). Bx; a; b) is the icomplete beta fuctio; cf. 86). Φx) is tabulated i 87). 3 Here, x / deotes the distributio media of the radom variable X.
10 x TABLE OF FORMULÆ, FMSN5/MASM3, Distributio Parameter restrictios Expectatio Variace 7) Gamma distributio, Gammaa; b) fx) = F x) = = b Γa) b x)a e b x ; x 0 = Γa;b x) Γa) ; x > 0 a > 0, b > 0 a b a b 7) Expoetial distributio, Expa) F x) = = e x/a ; x > 0 a > 0 a a 73) Gumbel type I extreme value) distributio e F x) = e x b)/a a > 0 b + γa a p 6 74) Fréchet type II extreme value) distributio 3, 4 F x) = = 0; x b x b = e a ) c ; x > b 75) x b Type III = e a ) c ; x < b F x) = extreme value = ; x b distributio 3, 5 76) Weibull distributio 3 F x) = = 0; x b x b = e a ) c ; x > b a > 0, c > 0 a > 0, c > 0 a > 0, c > 0 b + aγ» /c) a Γ c ) Γ «c ) b aγ +» /c) a Γ + c ) Γ + «c ) b + aγ +» /c) a Γ + c ) Γ + «c ) 77) Rayleigh distributio = 0; x b F x) = x b = e a ) ; x > b a > 0 b + a p a p 4 ) 78) Chi-square distributio 6, q ), Gamma ; ) fx) = = / Γ ) x/)/) e x/ ; x > 0 F x) = = Γ ; x ) Γ ) ; x > 0 Z 79) Studet s t-distributio 3, 7, t) fx) = + Γ ) p Γ ) + x )+)/ 8 >< = F x) = B +x ; ; ) ; x < 0 B; ; ) >: = B +x ; ; ) ; x 0 B; ; ) Z 0 Γa) is the gamma fuctio; cf. 84). Γa; b x) is the upper icomplete gamma fuctio; cf. 85). γ is Euler s costat. γ = lim k `P k i= k ) l k = 0, Γ is the gamma fucito; cf. 84). 4 Expectatio exists if ad oly if c >. Variace exists if ad oly if c >. 5 If X is a type III extreme value distributed radom variable, the X i.e. the egative of X) is Weibull distributed. Therefore the type III extreme value distributio ow ad the is called the extreme value distributio of Weibull type. 6 Γ ) is the gamma fuctio; cf. 84). Γ ; x ) is the upper icomplete gamma fuctio; cf. 85). 7 B is the icomplete beta fuctio; cf. 86). Variace exists if ad oly if 3.
11 TABLE OF FORMULÆ, FMSN5/MASM3, xi Distributio Parameter restrictios Expectatio Variace 80) Fisher s F-distributio, F ; ) 8) Pareto distributio, c > 0) fx) = = [x 0] = 0 = [x > 0] = = Γ + ) Γ ) Γ / / x ) + x) + )/ 8 < F x) = : = B + x ; ; ) ; x > 0 B; ; ) 8 >< F x) = = c x a >: )/c ; 0 < x < a c = ; x > a c Z Z a > 0, c > 0 a c + + ) ) 4) a c + )c + ) 8) Pareto distributio, c < 0) F x) = = + c x a ) / c ; x > 0 a > 0, c < 0 a c + a c + )c + ) 83) Pareto distributio 3, c = 0) F x) = = e x/a ; x > 0 a > 0 a a B is the icomplete beta fuctio; cf. 86). Expectatio exists if ad oly if 3. Variace exists if ad oly if 5. Expectatio exists if ad oly if c >. Variace exists if ad oly if c >. 3 This is a expoetial distributio, Expa). Some fuctios 84) The gamma fuctio is defied for p>0) by 84.) Γp) = 0 ξ p e ξ dξ, p>0 Some properties of the gamma fuctio: 84.) Γp) = p )!, p ; ; 3;...} 84.3) Γ ) = p 84.4) Γp + ) = p Γp); p>0 85) The icomplete gamma fuctio is defied for p>0, x>0) by 85.) Γp; x) = x ξ p e ξ dξ, x 0, p>0 A property of the icomplete gamma fuctio: 85.) Γp; 0) = Γp), p > 0 86) The icomplete beta fuctio is defied for a>0, b>0, 0 x ) by 86.) Bx; a; b) = x A property of the icomplete beta fuctio: 86.) B; a; b) = 0 ξ a ξ) b dξ, 0 x, a>0, b>0 Γa) Γb), a>0, b>0 Γa + b)
12 xii TABLE OF FORMULÆ, FMSN5/MASM3, Table of the stadard-ormal distributio fuctio 87) If X N0; ), the PX x) = Φx), where Φ ) is a o-elemetary fuctio give by Φx) = x p e ξ dξ. This table gives with 5 correct decimals) the fuctio values Φx) for x = 0,00:0,0:3,99. For egative values of x, use that Φ x) = Φx). 0,00 0,0 0,0 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,0 0, , , ,5 97 0, , ,53 9 0, , , , 0, , , ,55 7 0, , , , ,57 4 0, , 0, , , , , , , , ,60 6 0, ,3 0,67 9 0,6 7 0,65 5 0, , , , , , , ,4 0, , , , , , , , , , ,5 0, , , , , , ,7 6 0, , ,7 40 0,6 0, , , , , ,74 5 0, , , , ,7 0, ,76 5 0, , , , , , , , ,8 0, , , , , , ,805 0, , ,83 7 0,9 0, , ,8 0,83 8 0, , , , , ,838 9,0 0, , , , , , , , , ,86 4, 0, , , , , , , , , ,88 98, 0, , , , ,89 5 0, , , , ,90 47,3 0, , , , , ,9 49 0, , ,96 0,97 74,4 0,99 4 0, ,9 0 0, , , , ,99 0, ,93 89,5 0, , , , ,938 0, , , , ,944 08,6 0, , , , , , , , , ,954 49,7 0, , , , , , , , , ,963 7,8 0, , , , ,967 0, , , , ,970 6,9 0,97 8 0, , , , , , , , ,976 70,0 0, , , , , , , , ,98 4 0,98 69, 0,98 4 0, , , , ,984 0, , , ,985 74, 0, , , , , , , , , ,988 99,3 0, , , , , , , ,99 0, ,99 58,4 0, ,99 0 0,99 4 0, , , , , , ,993 6,5 0, , , , , , , , , ,995 0,6 0, , , , , , , ,996 0, ,996 43,7 0, , , , , , ,997 0, , ,997 36,8 0, , , , , , , , , ,998 07,9 0, , , , , , , , , , ,0 0, , , , , , , , , , , 0, , , , , , ,999 0, , , , 0, , , , , , , , , , ,3 0, , , , , , , , , , ,4 0, , , , , , , , , , ,5 0, , , , , , , , , , ,6 0, , , , , , , , , , ,7 0, , , , , , , , , , ,8 0, , , , , , , , , , ,9 0, , , , , , , , , ,999 97
13 TABLE OF FORMULÆ, FMSN5/MASM3, xiii Table of quatiles of Studet s t-distributio 88) If X t), the the α-quatile t α ) is defied by P X > t α ) ) = α, 0 < α < This table gives with 3 correct decimals) the α-quatile t α ) for α 0,; 0,05; 0,05; 0,0; 0,005; 0,00; 0,0005} ad for ::30; 40; 60; 0}. For values of α 0,9, use that t α ) = t α ), 0 < α < α 0, 0,05 0,05 0,0 0,005 0,00 0,0005 3,078 6,34,706 3,8 63,657 38, ,69,886,90 4,303 6,965 9,95,37 3,599 3,638,353 3,8 4,54 5,84 0,5,94 4,533,3,776 3,747 4,604 7,73 8,60 5,476,05,57 3,365 4,03 5,893 6,869 6,440,943,447 3,43 3,707 5,08 5,959 7,45,895,365,998 3,499 4,785 5,408 8,397,860,306,896 3,355 4,50 5,04 9,383,833,6,8 3,50 4,97 4,78 0,37,8,8,764 3,69 4,44 4,587,363,796,0,78 3,06 4,05 4,437,356,78,79,68 3,055 3,930 4,38 3,350,77,60,650 3,0 3,85 4, 4,345,76,45,64,977 3,787 4,40 5,34,753,3,60,947 3,733 4,073 6,337,746,0,583,9 3,686 4,05 7,333,740,0,567,898 3,646 3,965 8,330,734,0,55,878 3,60 3,9 9,38,79,093,539,86 3,579 3,883 0,35,75,086,58,845 3,55 3,850,33,7,080,58,83 3,57 3,89,3,77,074,508,89 3,505 3,79 3,39,74,069,500,807 3,485 3,768 4,38,7,064,49,797 3,467 3,745 5,36,708,060,485,787 3,450 3,75 6,35,706,056,479,779 3,435 3,707 7,34,703,05,473,77 3,4 3,690 8,33,70,048,467,763 3,408 3,674 9,3,699,045,46,756 3,396 3,659 30,30,697,04,457,750 3,385 3,646 40,303,684,0,43,704 3,307 3,55 60,96,67,000,390,660 3,3 3,460 0,89,658,980,358,67 3,60 3,373,8,645,960,36,576 3,090 3,9
14 xiv TABLE OF FORMULÆ, FMSN5/MASM3, Table of quatiles of the q distributio 89) If X q ), the the α-quatile χ α) is defied by P X > χ α) ) = α, 0 < α < This table gives the α-quatile χ α) for α 0,9995; 0,999; 0,99; 0,975; 0,95; 0,05; 0,05; 0,0; 0,005; 0,00; 0,0005} ad for ::30; 40:0:00}. α 0,9995 0,999 0,995 0,99 0,975 0,95 0,05 0,05 0,0 0,005 0,00 0,0005 < 0 < 0 < 0 < 0 3,84 5,04 6,635 7,879 0,83, < 0 < 0 0,000 0,00 0,0506 0,06 5,99 7,378 9,0 0,60 3,8 5,0 3 0,053 0,040 0,077 0,48 0,58 0,358 7,85 9,348,34,84 6,7 7,73 4 0,0639 0,0908 0,070 0,97 0,4844 0,707 9,488,4 3,8 4,86 8,47 0,00 5 0,58 0,0 0,47 0,5543 0,83,45,07,83 5,09 6,75 0,5, 6 0,994 0,38 0,6757 0,87,37,635,59 4,45 6,8 8,55,46 4,0 7 0,4849 0,5985 0,9893,39,690,67 4,07 6,0 8,48 0,8 4,3 6,0 8 0,704 0,857,344,646,80,733 5,5 7,53 0,09,95 6, 7,87 9 0,977,5,735,088,700 3,35 6,9 9,0,67 3,59 7,88 9,67 0,65,479,56,558 3,47 3,940 8,3 0,48 3, 5,9 9,59 3,4,587,834,603 3,053 3,86 4,575 9,68,9 4,7 6,76 3,6 33,4,934,4 3,074 3,57 4,404 5,6,03 3,34 6, 8,30 3,9 34,8 3,305,67 3,565 4,07 5,009 5,89,36 4,74 7,69 9,8 34,53 36,48 4,697 3,04 4,075 4,660 5,69 6,57 3,68 6, 9,4 3,3 36, 38, 5 3,08 3,483 4,60 5,9 6,6 7,6 5,00 7,49 30,58 3,80 37,70 39,7 6 3,536 3,94 5,4 5,8 6,908 7,96 6,30 8,85 3,00 34,7 39,5 4,3 7 3,980 4,46 5,697 6,408 7,564 8,67 7,59 30,9 33,4 35,7 40,79 4,88 8 4,439 4,905 6,65 7,05 8,3 9,390 8,87 3,53 34,8 37,6 4,3 44,43 9 4,9 5,407 6,844 7,633 8,907 0, 30,4 3,85 36,9 38,58 43,8 45,97 0 5,398 5,9 7,434 8,60 9,59 0,85 3,4 34,7 37,57 40,00 45,3 47,50 5,896 6,447 8,034 8,897 0,8,59 3,67 35,48 38,93 4,40 46,80 49,0 6,404 6,983 8,643 9,54 0,98,34 33,9 36,78 40,9 4,80 48,7 50,5 3 6,94 7,59 9,60 0,0,69 3,09 35,7 38,08 4,64 44,8 49,73 5,00 4 7,453 8,085 9,886 0,86,40 3,85 36,4 39,36 4,98 45,56 5,8 53,48 5 7,99 8,649 0,5,5 3, 4,6 37,65 40,65 44,3 46,93 5,6 54,95 6 8,538 9,,6,0 3,84 5,38 38,89 4,9 45,64 48,9 54,05 56,4 7 9,093 9,803,8,88 4,57 6,5 40, 43,9 46,96 49,64 55,48 57,86 8 9,656 0,39,46 3,56 5,3 6,93 4,34 44,46 48,8 50,99 56,89 59,30 9 0,3 0,99 3, 4,6 6,05 7,7 4,56 45,7 49,59 5,34 58,30 60, ,80,59 3,79 4,95 6,79 8,49 43,77 46,98 50,89 53,67 59,70 6,6 40 6,9 7,9 0,7,6 4,43 6,5 55,76 59,34 63,69 66,77 73,40 76, ,46 4,67 7,99 9,7 3,36 34,76 67,50 7,4 76,5 79,49 86,66 89, ,34 3,74 35,53 37,48 40,48 43,9 79,08 83,30 88,38 9,95 99,6 0, ,47 39,04 43,8 45,44 48,76 5,74 90,53 95,0 00,4 04,,3 5, ,79 46,5 5,7 53,54 57,5 60,39 0,9 06,6,3 6,3 4,8 8,3 90 5,8 54,6 59,0 6,75 65,65 69,3 3, 8, 4, 8,3 37, 40, ,90 6,9 67,33 70,06 74, 77,93 4,3 9,6 35,8 40, 49,4 53, This is versio
p n r.01.05.10.15.20.25.30.35.40.45.50.55.60.65.70.75.80.85.90.95
r r Table 4 Biomial Probability Distributio C, r p q This table shows the probability of r successes i idepedet trials, each with probability of success p. p r.01.05.10.15.0.5.30.35.40.45.50.55.60.65.70.75.80.85.90.95
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