MEI EXAMINATION FORMULAE AND TABLES (MF2)

MEI EXAMINATION FORMULAE AND TABLES (MF) For use with: Advaced Geeral Certificate of Educatio Advaced Subsidiary Geeral Certificate of Educatio MEI STRUCTURED MATHEMATICS ad Advaced Subsidiary GCE QUANTITATIVE METHODS (MEI) MF CST5

MEI STRUCTURED MATHEMATICS ad QUANTITATIVE METHODS (MEI) EXAMINATION FORMULAE AND TABLES

ALGEBRA Arithmetic series Geeral (kth) term, u k = a + (k )d last (th) term, l = u = a + ( l)d Sum to terms, S = (a + l) = [a + ( )d] Geometric series Geeral (kth) term, u k = a r k a( r Sum to terms, S = ) = r Sum to ifiity S = a, < r < r Biomial expasios Whe is a positive iteger (a + b) = a + ( ) a b + ( ) a b +... + ( ) a r b r +... b, where ( ) =! C r = ( ) ( ) ( ) + = + r r r + r + r!( r)! r a(r ) r Geeral case ( )! ( )... ( r +).... r ( + x) = + x + x +... + x r +..., x <, Logarithms ad expoetials e xl a = a x log a x = Numerical solutio of equatios Newto-Raphso iterative formula for solvig f(x) = 0, x + = x Complex Numbers {r(cos θ + j si θ)} = r (cos θ + j si θ) e jθ = cos θ + j si θ The roots of z = are give by z = exp( j) for k = 0,,,..., Fiite series r = ( + )( + ) 6 r= r= πk log b x log b a r 3 = 4 ( + ) f(x ) f'(x ) Ifiite series f(x) = f(0) + xf'(0) + f"(0) +... + f (r) (0) +... f(x) = f(a)+(x a)f'(a) + f"(a) +... + +... f(a + x) = f(a) + xf'(a) + f"(a) +... + f (r) (a) +... e x = exp(x) = + x + +... + +..., all x x l( + x) = x +... + ( ) r+ +..., < x x 3 3! si x = x +... + ( ) r +..., all x x! cos x = +... + ( ) r +..., all x x 3 3 arcta x = x +... + ( ) r +..., x x 3 3! sih x = x + + +... + +..., all x x! cosh x = + + +... + +..., all x x 3 3 x! x 3 3 x 5 5! x 4 4! x 5 5 x 5 5! x 4 4! x 5 5 x! x! (x a)! x r+ (r + )! x r (r)! x r (r)! x r+ r + x r+ (r +) x r+ (r + )! artah x = x + + +... + +..., < x < x r r! x r r x r r! x r r! (x a) r f (r) (a) r! Hyperbolic fuctios cosh x sih x =, sihx = sihx coshx, coshx = cosh x + sih x Matrices arsih x = l(x + ), arcosh x = l(x + x + x ), x + x x artah x = l ( ), x < cos θ si θ Aticlockwise rotatio through agle θ, cetre O: ( si θ cos θ ) ( ) Reflectio i the lie y = x ta θ : cos θ si θ si θ cos θ

TRIGONOMETRY, VECTORS AND GEOMETRY Cosie rule cos A = b + c a bc (etc.) a = b + c bc cos A (etc.) Trigoometry si (θ ± φ) = si θ cos φ ± cos θ si φ cos (θ ± φ) = cos θ cos φ si θ si φ ta θ ± ta φ ta (θ ± φ) =, [(θ ± φ) (k + )π] ta θ ta φ B c a A b C For t = ta θ : si θ = t, cos θ = ( t ) ( + t ) ( + t ) si θ + si φ = si (θ + φ) cos (θ φ) si θ si φ = cos (θ + φ) si (θ φ) cos θ + cos φ = cos (θ + φ) cos (θ φ) cos θ cos φ = si (θ + φ) si (θ φ) Vectors ad 3-D coordiate geometry (The positio vectors of poits A, B, C are a, b, c.) The positio vector of the poit dividig AB i the ratio λ:μ is μa + λb (λ + μ) Lie: Cartesia equatio of lie through A i directio u is y a z a 3 x a u = = ( = t ) u u 3 The resolved part of a i the directio u is a. u u Plae: Cartesia equatio of plae through A with ormal is x + y + 3 z + d = 0 where d = a. The plae through o-colliear poits A, B ad C has vector equatio r = a + s(b a) + t(c a) = ( s t) a + sb + tc The plae through A parallel to u ad v has equatio r = a + su + tv Perpedicular distace of a poit from a lie ad a plae Lie: (x,y ) from ax + by + c = 0 : ax + by + c a + b Plae: (α,β,γ) from x + y + 3 z + d = 0 : α + β + 3 γ + d ( + + 3 ) Vector product a b = a b siθ ^ i a b ( a b 3 a 3 b ) = j a b = a 3 b a b 3 k a 3 b 3 a b a b a b c a. (b c) = a b c a 3 b 3 c 3 = b. (c a) = c. (a b) a (b c) = (c. a) b (a. b) c Coics Rectagular Ellipse Parabola Hyperbola hyperbola Stadard x y + = = form y x = 4ax y xy = c a b a b Parametric form (acosθ, bsiθ) (at, at) (asecθ, btaθ) (ct, ) c Eccetricity e < b = a ( e ) e > e = b = a (e e = ) Foci (± ae, 0) (a, 0) (± ae, 0) (±c, ±c ) Directrices x = ± a e x = a x = ± a e x + y = ±c Asymptotes oe oe x y a = ± b x = 0, y = 0 t Ay of these coics ca be expressed i polar l coordiates (with the focus as the origi) as: r = + e cos θ where l is the legth of the semi-latus rectum. Mesuratio Sphere : Surface area = 4πr Coe : Curved surface area = πr slat height 3

CALCULUS Differetiatio f(x) f'(x) ta kx ksec kx sec x sec x ta x cot x cosec x cosec x cosec x cot x arcsi x ( x ) arccos x arcta x ( x ) + x sih x cosh x cosh x sih x tah x sech x arsih x arcosh x artah x ( + x ) (x ) ( x ) du v u u Quotiet rule y =, dy = dx v dx v dv dx Trapezium rule b ydx h{(y a 0 + y ) + (y + y +... + y Itegratio by parts u dx = uv v dx dv dx Area of a sector A = r dθ (polar coordiates) du dx )}, where h = A = (xẏ yẋ) dt (parametric form) Arc legth s = (ẋ + ẏ ) dt (parametric form) dy dx s = ( + [ ] ) dx (cartesia coordiates) s = dr (r + [ ] ) dθ (polar coordiates) dθ b a Itegratio f(x) f(x) dx (+ a costat) sec kx (l/k) ta kx ta x l sec x cot x l si x x cosec x l cosec x + cot x = l ta x x a a l x a x + a sec x l sec x + ta x = l ta ( + 4 π ) (a x ) a + x a x arcsi ( ), x < a arcta( a x ) sih x cosh x cosh x sih x tah x l cosh x (a + x ) (x a ) a a x a l a + x a x x a = artah ( ), x < a arsih ( ) or l (x + ), x a a arcosh ( ) or l (x + ), x > a, a > 0 x a x + a x a Surface area of revolutio S x = π y ds = π y (ẋ + ẏ ) dt Curvature dψ ds ẋ ÿ ẍ ẏ (ẋ + ẏ ) 3/ κ = = = S y = π x ds = π x (ẋ + ẏ ) dt d y dx dy ( + [ ] ) 3 / dx Radius of curvature ρ = κ, Cetre of curvature c = r + ρ ^ L'Hôpital s rule Lim x a f(x) g(x) If f(a) = g(a) = 0 ad g'(a) 0 the = f'(a) g'(a) Multi-variable calculus g/ x g/ y g/ z grad g = ( ) w x w y w z For w = g(x, y, z), δw = δx+ δy + δz 4

MECHANICS Cetre of mass (uiform bodies) Momets of iertia (uiform bodies, mass M) Triagular lamia: 3 alog media from vertex Thi rod, legth l, about perpedicular axis through cetre: 3 Ml Solid hemisphere of radius r: 3 8 r from cetre Rectagular lamia about axis i plae bisectig edges of legth l: 3 Ml Hemispherical shell of radius r: r from cetre 4 Thi rod, legth l, about perpedicular axis through ed: 3 Ml Solid coe or pyramid of height h: 4 h above the base o the lie from cetre of base to vertex Sector of circle, radius r, agle θ: r si θ from cetre 3θ Arc of circle, radius r, agle θ at cetre: 3 r si θ θ from cetre Coical shell, height h: h above the base o the lie from the cetre of base to the vertex 4 Rectagular lamia about edge perpedicular to edges of legth l: 3 Ml Rectagular lamia, sides a ad b, about perpedicular axis through cetre: 3 M(a + b ) Hoop or cylidrical shell of radius r about perpedicular axis through cetre: Mr Hoop of radius r about a diameter: Mr Disc or solid cylider of radius r about axis: Mr Disc of radius r about a diameter: 4 Mr Motio i polar coordiates Motio i a circle Trasverse velocity: v = rθ Radial acceleratio: rθ = Trasverse acceleratio: v = rθ v r Solid sphere of radius r about a diameter: 5 Mr Spherical shell of radius r about a diameter: 3 Mr Parallel axes theorem: I A = I G + M(AG) Perpedicular axes theorem: I z = I x + I y (for a lamia i the (x, y) plae) Geeral motio Radial velocity: ṙ Trasverse velocity: rθ Radial acceleratio: r rθ Trasverse acceleratio: rθ + ṙθ = r (r θ ) d dt Momets as vectors The momet about O of F actig at r is r F 5

STATISTICS Probability P(A B) = P(A) + P(B) P(A B) P(A B) = P(A). P(B A) P(A B) = P(B A)P(A) P(B A)P(A) + P(B A')P(A') Populatios Bayes Theorem: P(A j B) = P(A j )P(B A j ) P(A i )P(B A i ) Discrete distributios X is a radom variable takig values x i i a discrete distributio with P(X = x i ) = p i Expectatio: μ = E(X) = x i p i Variace: σ = Var(X) = (x i μ) p i = x i pi μ For a fuctio g(x): E[g(X)] = g(x i )p i Cotiuous distributios X is a cotiuous variable with probability desity fuctio (p.d.f.) f(x) Expectatio: μ = E(X) = x f(x)dx Variace: σ = Var(X) = (x μ) f(x)dx = x f(x)dx μ For a fuctio g(x): E[g(X)] = g(x)f(x)dx Cumulative distributio fuctio F(x) = P(X x) = x f(t)dt Correlatio ad regressio For a sample of pairs of observatios (x i, y i ) ( x i ) S xx = (x i x ) = x i, S yy = (y i y ) = y i, S xy = (x i x )(y i y ) = x i y i ( x i )( y i ) ( y i ) S xy Covariace = ( )( ) = i i i i x x y y xy xy Product-momet correlatio: Pearso s coefficiet S xy Σ ( xi x)( yi y) r = = = Sxx Syy [ Σ( xi x) Σ( yi y) ] x xy i i x xy y y i i Rak correlatio: Spearma s coefficiet r s = 6 d i ( ) Regressio Least squares regressio lie of y o x: y y = b(x x ) xy i i S b = xy (x = i x ) (y i y ) xy = S (x i ) xx x x x Estimates Ubiased estimates from a sigle sample X for populatio mea μ; Var X = σ S for populatio variace σ where S = (x i x ) f i Probability geeratig fuctios For a discrete distributio G(t) = E(t X ) E(X) = G'(); Var(X) = G"() + μ μ G X + Y (t) = G X (t) G Y (t) for idepedet X, Y Momet geeratig fuctios: M X (θ) = E(e θx ) E(X) = M'(0) = μ; E(X ) = M () (0) Var(X) = M"(0) {M'(0)} M X + Y (θ) = M X (θ) M Y (θ) for idepedet X, Y i 6

STATISTICS Markov Chais p + = p P Log ru proportio p = pp Bivariate distributios Covariace Cov(X, Y) = E[(X μ X )(Y μ Y )] = E(XY) μ X μ Y Cov(X, Product-momet correlatio coefficiet ρ = Y) σ X σ Y Sum ad differece Var(aX ± by) = a Var(X) + b Var(Y) ± ab Cov (X,Y) If X, Y are idepedet: Var(aX ± by) = a Var(X) + b Var(Y) E(XY) = E(X) E(Y) Codig X = ax' + b } Cov(X, Y) = ac Cov(X', Y') Y = cy' + d Aalysis of variace Oe-factor model: x ij = μ + α i + ε ij, where ε ij ~ N(0,σ ) SS B = i ( i ) x x = T i i i i T SS T = (x ij x ) = x ij T i j i j Regressio Y i α + βx i + ε i α + βf(x i ) + ε i α + βx i + γz i + ε i ε i ~ N(0, σ ) a, b, c are estimates for α, β, γ. For the model Y i = α + βx i + ε i, S xy S xx RSS (y i a bx i ) (y i a bf(x i )) (y i a bx i cz i ) σ S xx b =, b ~ N( β, ), ~ t a = b, a ~ N( α, σ x y x i S xx ) a + bx 0 ~ N(α + βx 0, σ { + (x 0 ) } (S xy ) RSS = S yy = S yy ( r ) S xx b β σˆ / S xx x S xx No. of parameters, p 3 ^σ = RSS p Radomised respose techique y E(^p) = ( θ) Var(^p) = (θ ) [(θ ) p + ( θ)][θ (θ )p] (θ ) Factorial desig Iteractio betwee st ad d of 3 treatmets (Abc abc) + (AbC abc) (ABc abc) + (ABC abc) ( ) { } Expoetial smoothig ^y+ = αy + α( α)y + α( α) y +... + α( α) y + ( α) y 0 ^y+ = ^y + α(y ^y ) ^y+ = αy + ( α) ^y 7

STATISTICS: HYPOTHESIS TESTS Descriptio Pearso s product momet correlatio test Spearma rak correlatio test Normal test for a mea t-test for a mea χ test t-test for paired sample Normal test for the differece i the meas of samples with differet variaces r = Test statistic Distributio x xy i i xy y x i i y r s = 6 d i ( ) x μ N(0, ) σ / x μ s/ t ( f f ) o e f e χ v ( ) μ x x s / t with ( ) degrees of freedom ( x y) ( μ μ ) σ σ + N(0, ) Descriptio Test statistic Distributio t-test for the differece i the meas of samples where s = ( x y) ( μ μ) s + ( )s + ( )s + t + Wilcoxo sigle sample test A statistic T is calculated from the raked data. See tables Wilcoxo Rak-sum (or Ma-Whitey) -Sample test Samples size m, : m Wilcoxo W = sum of raks of sample size m Ma-Whitey T = W m(m + ) See tables Normal test o biomial proportio p θ θ( θ) N(0, ) χ test for variace ( ) σ s χ F-test o ratio of two variaces s /σ s /σ, s > s F, 8

STATISTICS: DISTRIBUTIONS Name Biomial B(, p) Discrete Poisso (λ) Discrete Normal N(μ, σ ) Cotiuous Uiform (Rectagular) o [a, b] Cotiuous Expoetial Cotiuous Geometric Discrete Negative biomial Discrete Fuctio P(X = r) = C r q r p r, for r = 0,,...,, 0 < p <, q = p P(X = r) = e λ λ r r!, for r = 0,,..., λ > 0 f(x) = exp ( ( ) ), σ π x μ σ < x < f(x) =, a x b b a f(x) = λe λx, x 0, λ >0 P(X = r) = q r p, r =,,..., 0 < p <, q = p P(X = r) = r C q r p, r =, +,..., 0 < p <, q = p Mea Variace p pq λ λ μ σ a + b (b a) λ λ q p p p q p p.g.f. G(t) (discrete) m.g.f. M(θ) (cotiuous) G(t) = (q + pt) G(t) = e λ(t ) M(θ) = exp(μθ + σ θ ) M(θ) = e bθ e aθ (b a)θ M(θ) = λ λ θ G(t) = pt qt pt qt G(t) = ( ) 9

NUMERICAL ANALYSIS DECISION & DISCRETE MATHEMATICS Numerical Solutio of Equatios The Newto-Raphso iteratio for solvig f(x) = 0 : x + = x Numerical itegratio The trapezium rule b ydx h{(y a 0 + y ) + (y + y +... + y )}, where h = f(x ) f'(x ) b a The mid-ordiate rule b ydx h(y + y a +... + y + y ), where h = b a Simpso s rule for eve b ydx 3 h{(y a 0 + y ) + 4(y + y 3 +... + y ) + (y + y 4 +... + y )}, where h = b a The Gaussia -poit itegratio rule h f ( ) h h x dx h + h f f 3 3 Iterpolatio/fiite differeces Lagrage s polyomial : P (x) = L r (x)f(x r ) where L r (x) = Newto s forward differece iterpolatio formula (x x 0 ) h (x x 0 )(x x )!h f(x) = f(x 0 ) + Δf(x 0 ) + Δ f(x 0 ) +... i=0 i r x x i x r x i Newto s divided differece iterpolatio formula f(x) = f[x 0 ] + (x x 0 )f[x 0, x ] + (x x 0 ) (x x )f[x 0, x, x ] +... Numerical differetiatio f"(x) f(x + h) f(x) + f(x h) h Taylor polyomials f(a + h) = f(a) + hf'(a) + h! f"(a) + error f(a + h) = f(a) + hf'(a) + h f"(a + ξ), 0 < ξ < h! (x f(x) = f(a) + (x a)f '(a) + a) f"(a) + error! (x f(x) = f(a) + (x a)f '(a) + a) f"(η), a < η < x! Numerical solutio of differetial equatios dy For = f(x, y): dx Euler s method : y r + = y r + hf(x r, y r ); x r+ = x r + h Ruge-Kutta method (order ) (modified Euler method) y r + = y r + (k + k ) where k = h f(x r, y r ), k = h f(x r + h, y r + k ) Ruge-Kutta method, order 4: y r+ = y r + 6 (k + k + k 3 + k 4 ), where k = hf(x r, y r ) k = hf(x r + h, y r + k ) k 3 = hf(x r + h, y r + k ) k 4 = hf(x r + h, y r + k 3 ). Logic gates NOT AND OR NAND 0

Statistical Tables 7 Cumulative biomial probability 8 0 Cumulative Poisso probability Critical values for correlatio coefficiets The Normal distributio ad its iverse 3 Percetage poits of the χ distributio 3 Percetage poits of the t-distributio 4 5 Critical values for the F-test 6 7 Critical values for the Ma-Whitey test 8 9 Critical values for the Wilcoxo Rak Sum -sample test 30 Critical values for the Wilcoxo Sigle sample ad Paired sample tests 30 Shewhart Chart: Actio ad Warig lies 3 Estimatio of stadard deviatio from rage 3 3 Radom permutatios

CUMULATIVE BINOMIAL PROBABILITY The Biomial distributio: cumulative probabilites P(X x) = x r=0 Cr ( p) r p r x p 0.050 0.00 0.50 /6 0.00 0.50 0.300 /3 0.350 0.400 0.450 0.500 0.550 0.600 0.650 /3 0 0.9500 0.9000 0.8500 0.8333 0.8000 0.7500 0.7000 0.6667 0.6500 0.6000 0.5500 0.5000 0.4500 0.4000 0.3500 0.3333.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0 0.905 0.800 0.75 0.6944 0.6400 0.565 0.4900 0.4444 0.45 0.3600 0.305 0.500 0.05 0.600 0.5 0. 0.9975 0.9900 0.9775 0.97 0.9600 0.9375 0.900 0.8889 0.8775 0.8400 0.7975 0.7500 0.6975 0.6400 0.5775 0.5556.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 3 0 0.8574 0.790 0.64 0.5787 0.50 0.49 0.3430 0.963 0.746 0.60 0.664 0.50 0.09 0.0640 0.049 0.0370 0.998 0.970 0.939 0.959 0.8960 0.8437 0.7840 0.7407 0.783 0.6480 0.5748 0.5000 0.45 0.350 0.88 0.593 0.9999 0.9990 0.9966 0.9954 0.990 0.9844 0.9730 0.9630 0.957 0.9360 0.9089 0.8750 0.8336 0.7840 0.754 0.7037 3.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 4 0 0.845 0.656 0.50 0.483 0.4096 0.364 0.40 0.975 0.785 0.96 0.095 0.065 0.040 0.056 0.050 0.03 0.9860 0.9477 0.8905 0.868 0.89 0.7383 0.657 0.596 0.5630 0.475 0.390 0.35 0.45 0.79 0.65 0. 0.9995 0.9963 0.9880 0.9838 0.978 0.949 0.963 0.8889 0.8735 0.808 0.7585 0.6875 0.6090 0.548 0.4370 0.4074 3.0000 0.9999 0.9995 0.999 0.9984 0.996 0.999 0.9877 0.9850 0.9744 0.9590 0.9375 0.9085 0.8704 0.85 0.805 4.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 5 0 0.7738 0.5905 0.4437 0.409 0.377 0.373 0.68 0.37 0.60 0.0778 0.0503 0.033 0.085 0.00 0.0053 0.004 0.9774 0.985 0.835 0.8038 0.7373 0.638 0.58 0.4609 0.484 0.3370 0.56 0.875 0.3 0.0870 0.0540 0.0453 0.9988 0.994 0.9734 0.9645 0.94 0.8965 0.8369 0.790 0.7648 0.686 0.593 0.5000 0.4069 0.374 0.35 0.099 3.0000 0.9995 0.9978 0.9967 0.9933 0.9844 0.969 0.9547 0.9460 0.930 0.8688 0.85 0.7438 0.6630 0.576 0.539 4.0000 0.9999 0.9999 0.9997 0.9990 0.9976 0.9959 0.9947 0.9898 0.985 0.9688 0.9497 0.9 0.8840 0.8683 5.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 6 0 0.735 0.534 0.377 0.3349 0.6 0.780 0.76 0.0878 0.0754 0.0467 0.077 0.056 0.0083 0.004 0.008 0.004 0.967 0.8857 0.7765 0.7368 0.6554 0.5339 0.40 0.35 0.39 0.333 0.636 0.094 0.069 0.040 0.03 0.078 0.9978 0.984 0.957 0.9377 0.90 0.8306 0.7443 0.6804 0.647 0.5443 0.445 0.3438 0.553 0.79 0.74 0.00 3 0.9999 0.9987 0.994 0.993 0.9830 0.964 0.995 0.8999 0.886 0.808 0.7447 0.6563 0.5585 0.4557 0.359 0.396 4.0000 0.9999 0.9996 0.9993 0.9984 0.9954 0.989 0.98 0.9777 0.9590 0.9308 0.8906 0.8364 0.7667 0.6809 0.6488 5.0000.0000.0000 0.9999 0.9998 0.9993 0.9986 0.998 0.9959 0.997 0.9844 0.973 0.9533 0.946 0.9 6.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 7 0 0.6983 0.4783 0.306 0.79 0.097 0.335 0.084 0.0585 0.0490 0.080 0.05 0.0078 0.0037 0.006 0.0006 0.0005 0.9556 0.8503 0.766 0.6698 0.5767 0.4449 0.394 0.634 0.338 0.586 0.04 0.065 0.0357 0.088 0.0090 0.0069 0.996 0.9743 0.96 0.904 0.850 0.7564 0.647 0.5706 0.533 0.499 0.364 0.66 0.59 0.0963 0.0556 0.0453 3 0.9998 0.9973 0.9879 0.984 0.9667 0.994 0.8740 0.867 0.800 0.70 0.6083 0.5000 0.397 0.898 0.998 0.733 4.0000 0.9998 0.9988 0.9980 0.9953 0.987 0.97 0.9547 0.9444 0.9037 0.847 0.7734 0.6836 0.580 0.4677 0.494 5.0000 0.9999 0.9999 0.9996 0.9987 0.996 0.993 0.990 0.98 0.9643 0.9375 0.8976 0.844 0.766 0.7366 6.0000.0000.0000 0.9999 0.9998 0.9995 0.9994 0.9984 0.9963 0.99 0.9848 0.970 0.950 0.945 7.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0.700 0.750 0.800 5/6 0.850 0.900 0.950 0.3000 0.500 0.000 0.667 0.500 0.000 0.0500.0000.0000.0000.0000.0000.0000.0000 0.0900 0.065 0.0400 0.078 0.05 0.000 0.005 0.500 0.4375 0.3600 0.3056 0.775 0.900 0.0975.0000.0000.0000.0000.0000.0000.0000 0.070 0.056 0.0080 0.0046 0.0034 0.000 0.000 0.60 0.563 0.040 0.074 0.0608 0.080 0.0073 0.6570 0.578 0.4880 0.43 0.3859 0.70 0.46.0000.0000.0000.0000.0000.0000.0000 0.008 0.0039 0.006 0.0008 0.0005 0.000 0.0000 0.0837 0.0508 0.07 0.06 0.00 0.0037 0.0005 0.3483 0.67 0.808 0.39 0.095 0.053 0.040 0.7599 0.6836 0.5904 0.577 0.4780 0.3439 0.855.0000.0000.0000.0000.0000.0000.0000 0.004 0.000 0.0003 0.000 0.000 0.0000 0.0308 0.056 0.0067 0.0033 0.00 0.0005 0.0000 0.63 0.035 0.0579 0.0355 0.066 0.0086 0.00 0.478 0.367 0.67 0.96 0.648 0.085 0.06 0.839 0.767 0.673 0.598 0.5563 0.4095 0.6.0000.0000.0000.0000.0000.0000.0000 0.0007 0.000 0.000 0.0000 0.0000 0.0000 0.009 0.0046 0.006 0.0007 0.0004 0.000 0.0000 0.0705 0.0376 0.070 0.0087 0.0059 0.003 0.000 0.557 0.694 0.0989 0.063 0.0473 0.059 0.00 0.5798 0.466 0.3446 0.63 0.35 0.43 0.038 0.884 0.80 0.7379 0.665 0.69 0.4686 0.649.0000.0000.0000.0000.0000.0000.0000 0.000 0.000 0.0000 0.0000 0.0000 0.0038 0.003 0.0004 0.000 0.000 0.0000 0.088 0.09 0.0047 0.000 0.00 0.000 0.0000 0.60 0.0706 0.0333 0.076 0.0 0.007 0.000 0.359 0.436 0.480 0.0958 0.0738 0.057 0.0038 0.6706 0.555 0.433 0.330 0.834 0.497 0.0444 0.976 0.8665 0.7903 0.709 0.6794 0.57 0.307.0000.0000.0000.0000.0000.0000.0000

CUMULATIVE BINOMIAL PROBABILITY x p 0.050 0.00 0.50 /6 0.00 0.50 0.300 /3 0.350 0.400 0.450 0.500 0.550 0.600 0.650 /3 0.700 0.750 0.800 5/6 0.850 0.900 0.950 8 0 0.6634 0.4305 0.75 0.36 0.678 0.00 0.0576 0.0390 0.039 0.068 0.0084 0.0039 0.007 0.0007 0.000 0.000 0.000 0.0000 0.0000 0.948 0.83 0.657 0.6047 0.5033 0.367 0.553 0.95 0.69 0.064 0.063 0.035 0.08 0.0085 0.0036 0.006 0.003 0.0004 0.000 0.0000 0.0000 0.994 0.969 0.8948 0.865 0.7969 0.6785 0.558 0.468 0.478 0.354 0.0 0.445 0.0885 0.0498 0.053 0.097 0.03 0.004 0.00 0.0004 0.000 0.0000 3 0.9996 0.9950 0.9786 0.9693 0.9437 0.886 0.8059 0.744 0.7064 0.594 0.4770 0.3633 0.604 0.737 0.06 0.0879 0.0580 0.073 0.004 0.0046 0.009 0.0004 0.0000 4.0000 0.9996 0.997 0.9954 0.9896 0.977 0.940 0.9 0.8939 0.863 0.7396 0.6367 0.530 0.4059 0.936 0.587 0.94 0.38 0.0563 0.0307 0.04 0.0050 0.0004 5.0000 0.9998 0.9996 0.9988 0.9958 0.9887 0.9803 0.9747 0.950 0.95 0.8555 0.7799 0.6846 0.57 0.538 0.448 0.35 0.03 0.348 0.05 0.038 0.0058 6.0000.0000 0.9999 0.9996 0.9987 0.9974 0.9964 0.995 0.989 0.9648 0.9368 0.8936 0.8309 0.8049 0.7447 0.639 0.4967 0.3953 0.348 0.869 0.057 7.0000.0000 0.9999 0.9998 0.9998 0.9993 0.9983 0.996 0.996 0.983 0.968 0.960 0.944 0.8999 0.83 0.7674 0.775 0.5695 0.3366 8.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 9 0 0.630 0.3874 0.36 0.938 0.34 0.075 0.0404 0.060 0.007 0.00 0.0046 0.000 0.0008 0.0003 0.000 0.000 0.0000 00000 0.988 0.7748 0.5995 0.547 0.436 0.3003 0.960 0.43 0. 0.0705 0.0385 0.095 0.009 0.0038 0.004 0.000 0.0004 0.000 0.0000 0.0000 0.996 0.9470 0.859 0.87 0.738 0.6007 0.468 0.377 0.3373 0.38 0.495 0.0898 0.0498 0.050 0.0 0.0083 0.0043 0.003 0.0003 0.000 0.0000 0.0000 3 0.9994 0.997 0.966 0.950 0.944 0.8343 0.797 0.6503 0.6089 0.486 0.364 0.539 0.658 0.0994 0.0536 0.044 0.053 0.000 0.003 0.00 0.0006 0.000 4.0000 0.999 0.9944 0.99 0.9804 0.95 0.90 0.855 0.883 0.7334 0.64 0.5000 0.3786 0.666 0.77 0.448 0.0988 0.0489 0.096 0.0090 0.0056 0.0009 0.0000 5 0.9999 0.9994 0.9989 0.9969 0.9900 0.9747 0.9576 0.9464 0.9006 0.834 0.746 0.6386 0.574 0.39 0.3497 0.703 0.657 0.0856 0.0480 0.0339 0.0083 0.0006 6.0000.0000 0.9999 0.9997 0.9987 0.9957 0.997 0.9888 0.9750 0.950 0.90 0.8505 0.768 0.667 0.68 0.537 0.3993 0.68 0.783 0.409 0.0530 0.0084 7.0000.0000 0.9999 0.9996 0.9990 0.9986 0.996 0.9909 0.9805 0.965 0.995 0.8789 0.8569 0.8040 0.6997 0.5638 0.4573 0.4005 0.5 0.07 8.0000.0000 0.9999 0.9999 0.9997 0.999 0.9980 09954 0.9899 0.9793 0.9740 0.9596 0.949 0.8658 0.806 0.7684 0.66 0.3698 9.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0 0 0.5987 0.3487 0.969 0.65 0.074 0.0563 0.08 0.073 0.035 0.0060 0.005 0.000 0.0003 0.000 0.0000 0.0000 0.0000 0.939 0.736 0.5443 0.4845 0.3758 0.440 0.493 0.040 0.0860 0.0464 0.033 0.007 0.0045 0.007 0.0005 0.0004 0.000 0.0000 0.0000 0.9885 0.998 0.80 0.775 0.6778 0.556 0.388 0.99 0.66 0.673 0.0996 0.0547 0.074 0.03 0.0048 0.0034 0.006 0.0004 0.000 0.0000 0.0000 3 0.9990 0.987 0.9500 0.9303 0.879 0.7759 0.6496 0.5593 0.538 0.383 0.660 0.79 0.00 0.0548 0.060 0.097 0.006 0.0035 0.0009 0.0003 0.000 0.0000 4 0.9999 0.9984 0.990 0.9845 0.967 0.99 0.8497 0.7869 0.755 0.633 0.5044 0.3770 0.66 0.66 0.0949 0.0766 0.0473 0.097 0.0064 0.004 0.004 0.000 0.0000 5.0000 0.9999 0.9986 0.9976 0.9936 0.9803 0.957 0.934 0.905 0.8338 0.7384 0.630 0.4956 0.3669 0.485 0.3 0.503 0.078 0.038 0.055 0.0099 0.006 0.000 6.0000 0.9999 0.9997 0.999 0.9965 0.9894 0.9803 0.9740 0.945 0.8980 0.88 0.7340 0.677 0.486 0.4407 0.3504 0.4 0.09 0.0697 0.0500 0.08 0.000 7.0000.0000 0.9999 0.9996 0.9984 0.9966 0.995 0.9877 0.976 0.9453 0.9004 0.837 0.7384 0.7009 0.67 0.4744 0.3 0.48 0.798 0.070 0.05 8.0000.0000 0.9999 0.9996 0.9995 0.9983 0.9955 0.9893 0.9767 0.9536 0.940 0.8960 0.8507 0.7560 0.64 0.555 0.4557 0.639 0.086 9.0000.0000.0000 0.9999 0.9997 0.9990 0.9975 0.9940 0.9865 0.987 0.978 0.9437 0.896 0.8385 0.803 0.653 0.403 0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 0 0.5688 0.338 0.673 0.346 0.0859 0.04 0.098 0.06 0.0088 0.0036 0.004 0.0005 0.000 0.0000 0.0000 0.0000 0.898 0.6974 0.49 0.4307 0.3 0.97 0.30 0.075 0.0606 0.030 0.039 0.0059 0.00 0.0007 0.000 0.000 0.0000 0.0000 0.9848 0.904 0.7788 0.768 0.674 0.455 0.37 0.34 0.00 0.89 0.065 0.037 0.048 0.0059 0.000 0.004 0.0006 0.000 0.0000 0.0000 3 0.9984 0.985 0.9306 0.9044 0.8389 0.733 0.5696 0.476 0.456 0.963 0.9 0.33 0.060 0.093 0.0 0.0088 0.0043 0.00 0.000 0.000 0.0000 4 0.9999 0.997 0.984 0.9755 0.9496 0.8854 0.7897 0.70 0.6683 0.538 0.397 0.744 0.738 0.0994 0.050 0.0386 0.06 0.0076 0.000 0.0006 0.0003 0.0000 5.0000 0.9997 0.9973 0.9954 0.9883 0.9657 0.98 0.8779 0.853 0.7535 0.633 0.5000 0.3669 0.465 0.487 0. 0.078 0.0343 0.07 0.0046 0.007 0.0003 0.0000 6.0000 0.9997 0.9994 0.9980 0.994 0.9784 0.964 0.9499 0.9006 0.86 0.756 0.609 0.467 0.337 0.890 0.03 0.46 0.0504 0.045 0.059 0.008 0.000 7.0000 0.9999 0.9998 0.9988 0.9957 0.99 0.9878 0.9707 0.9390 0.8867 0.8089 0.7037 0.5744 0.574 0.4304 0.867 0.6 0.0956 0.0694 0.085 0.006 8.0000.0000 0.9999 0.9994 0.9986 0.9980 0.994 0.985 0.9673 0.9348 0.88 0.7999 0.7659 0.6873 0.5448 0.386 0.73 0. 0.0896 0.05 9.0000.0000 0.9999 0.9998 0.9993 0.9978 0.994 0.986 0.9698 0.9394 0.949 0.8870 0.809 0.6779 0.5693 0.5078 0.306 0.09 0.0000.0000.0000 0.9998 0.9995 0.9986 0.9964 0.99 0.9884 0.980 0.9578 0.94 0.8654 0.837 0.686 0.43.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 3

CUMULATIVE BINOMIAL PROBABILITY x p 0.050 0.00 0.50 /6 0.00 0.50 0.300 /3 0.350 0.400 0.450 0.500 0.550 0.600 0.650 /3 0.700 0.750 0.800 5/6 0.850 0.900 0.950 0 0.5404 0.84 0.4 0. 0.0687 0.037 0.038 0.0077 0.0057 0.00 0.0008 0.000 0.000 0.0000 0.0000 0.886 0.6590 0.4435 0.383 0.749 0.584 0.0850 0.0540 0.044 0.096 0.0083 0.003 0.00 0.0003 0.000 0.0000 0.0000 0.9804 0.889 0.7358 0.6774 0.5583 0.3907 0.58 0.8 0.53 0.0834 0.04 0.093 0.0079 0.008 0.0008 0.0005 0.000 0.0000 0.0000 3 0.9978 0.9744 0.9078 0.8748 0.7946 0.6488 0.495 0.393 0.3467 0.53 0.345 0.0730 0.0356 0.053 0.0056 0.0039 0.007 0.0004 0.000 0.0000 0.0000 4 0.9998 0.9957 0.976 0.9637 0.974 0.844 0.737 0.635 0.5833 0.438 0.3044 0.938 0.7 0.0573 0.055 0.088 0.0095 0.008 0.0006 0.000 0.000 0.0000 5.0000 0.9995 0.9954 0.99 0.9806 0.9456 0.88 0.83 0.7873 0.665 0.569 0.387 0.607 0.58 0.0846 0.0664 0.0386 0.043 0.0039 0.003 0.0007 0.000 6 0.9999 0.9993 0.9987 0.996 0.9857 0.964 0.9336 0.954 0.848 0.7393 0.68 0.473 0.3348 0.7 0.777 0.78 0.0544 0.094 0.0079 0.0046 0.0005 0.0000 7.0000 0.9999 0.9998 0.9994 0.997 0.9905 0.98 0.9745 0.947 0.8883 0.806 0.6956 0.568 0.467 0.3685 0.763 0.576 0.076 0.0364 0.039 0.0043 0.000 8.0000.0000 0.9999 0.9996 0.9983 0.996 0.9944 0.9847 0.9644 0.970 0.8655 0.7747 0.6533 0.6069 0.5075 0.35 0.054 0.5 0.09 0.056 0.00 9.0000.0000 0.9998 0.9995 0.999 0.997 0.99 0.9807 0.9579 0.966 0.8487 0.889 0.747 0.6093 0.447 0.36 0.64 0.09 0.096 0.0000.0000 0.9999 0.9997 0.9989 0.9968 0.997 0.9804 0.9576 0.9460 0.950 0.846 0.75 0.687 0.5565 0.340 0.84.0000.0000 0.9999 0.9998 0.999 0.9978 0.9943 0.993 0.986 0.9683 0.933 0.8878 0.8578 0.776 0.4596.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 3 0 0.533 0.54 0.09 0.0935 0.0550 0.038 0.0097 0.005 0.0037 0.003 0.0004 0.000 0.0000 0.0000 0.8646 0.63 0.3983 0.3365 0.336 0.67 0.0637 0.0385 0.096 0.06 0.0049 0.007 0.0005 0.000 0.0000 0.0000 0.0000 0.9755 0.866 0.690 0.68 0.507 0.336 0.05 0.387 0.3 0.0579 0.069 0.0 0.004 0.003 0.0003 0.000 0.000 0.0000 3 0.9969 0.9658 0.880 0.849 0.7473 0.5843 0.406 0.34 0.783 0.686 0.099 0.046 0.003 0.0078 0.005 0.006 0.0007 0.000 0.0000 4 0.9997 0.9935 0.9658 0.9488 0.9009 0.7940 0.6543 0.550 0.5005 0.3530 0.79 0.334 0.0698 0.03 0.06 0.0088 0.0040 0.000 0.000 0.0000 0.0000 5.0000 0.999 0.995 0.9873 0.9700 0.998 0.8346 0.7587 0.759 0.5744 0.468 0.905 0.788 0.0977 0.046 0.0347 0.08 0.0056 0.00 0.0003 0.000 0.0000 6 0.9999 0.9987 0.9976 0.9930 0.9757 0.9376 0.8965 0.8705 0.77 0.6437 0.5000 0.3563 0.88 0.95 0.035 0.064 0.043 0.0070 0.004 0.003 0.000 7.0000 0.9998 0.9997 0.9988 0.9944 0.988 0.9653 0.9538 0.903 0.8 0.7095 0.573 0.456 0.84 0.43 0.654 0.080 0.0300 0.07 0.0075 0.0009 0.0000 8.0000.0000 0.9998 0.9990 0.9960 0.99 0.9874 0.9679 0.930 0.8666 0.77 0.6470 0.4995 0.4480 0.3457 0.060 0.099 0.05 0.034 0.0065 0.0003 9.0000 0.9999 0.9993 0.9984 0.9975 0.99 0.9797 0.9539 0.907 0.834 0.77 0.6776 0.5794 0.457 0.57 0.58 0.80 0.034 0.003 0.0000 0.9999 0.9998 0.9997 0.9987 0.9959 0.9888 0.973 0.94 0.8868 0.863 0.7975 0.6674 0.4983 0.379 0.3080 0.339 0.045.0000.0000.0000 0.9999 0.9995 0.9983 0.995 0.9874 0.9704 0.965 0.9363 0.8733 0.7664 0.6635 0.607 0.3787 0.354.0000.0000 0.9999 0.9996 0.9987 0.9963 0.9949 0.9903 0.976 0.9450 0.9065 0.879 0.7458 0.4867 3.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 4 0 0.4877 0.88 0.08 0.0779 0.0440 0.078 0.0068 0.0034 0.004 0.0008 0.000 0.000 0.0000 0.0000 0.8470 0.5846 0.3567 0.960 0.979 0.00 0.0475 0.074 0.005 0.008 0.009 0.0009 0.0003 0.000 0.0000 0.0000 0.9699 0.846 0.6479 0.5795 0.448 0.8 0.608 0.053 0.0839 0.0398 0.070 0.0065 0.00 0.0006 0.000 0.000 0.0000 3 0.9958 0.9559 0.8535 0.8063 0.698 0.53 0.355 0.6 0.05 0.43 0.063 0.087 0.04 0.0039 0.00 0.0007 0.000 0.0000 4 0.9996 0.9908 0.9533 0.930 0.870 0.745 0.584 0.4755 0.47 0.793 0.67 0.0898 0.046 0.075 0.0060 0.0040 0.007 0.0003 0.0000 0.0000 5.0000 0.9985 0.9885 0.9809 0.956 0.8883 0.7805 0.6898 0.6405 0.4859 0.3373 0.0 0.89 0.0583 0.043 0.074 0.0083 0.00 0.0004 0.000 0.0000 6 0.9998 0.9978 0.9959 0.9884 0.967 0.9067 0.8505 0.864 0.695 0.546 0.3953 0.586 0.50 0.0753 0.0576 0.035 0.003 0.004 0.0007 0.0003 0.0000 7.0000 0.9997 0.9993 0.9976 0.9897 0.9685 0.944 0.947 0.8499 0.744 0.6047 0.4539 0.3075 0.836 0.495 0.0933 0.0383 0.06 0.004 0.00 0.000 8.0000 0.9999 0.9996 0.9978 0.997 0.986 0.9757 0.947 0.88 0.7880 0.667 0.54 0.3595 0.30 0.95 0.7 0.0439 0.09 0.05 0.005 0.0000 9.0000.0000 0.9997 0.9983 0.9960 0.9940 0.985 0.9574 0.90 0.838 0.707 0.5773 0.545 0.458 0.585 0.98 0.0690 0.0467 0.009 0.0004 0.0000 0.9998 0.9993 0.9989 0.996 0.9886 0.973 0.9368 0.8757 0.7795 0.7388 0.6448 0.4787 0.308 0.937 0.465 0.044 0.004.0000 0.9999 0.9999 0.9994 0.9978 0.9935 0.9830 0.960 0.96 0.8947 0.839 0.789 0.559 0.405 0.35 0.584 0.030.0000.0000 0.9999 0.9997 0.999 0.997 0.999 0.9795 0.976 0.955 0.8990 0.80 0.7040 0.6433 0.454 0.530 3.0000.0000 0.9999 0.9998 0.999 0.9976 0.9966 0.993 0.98 0.9560 0.9 0.897 0.77 0.53 4.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 4

CUMULATIVE BINOMIAL PROBABILITY x p 0.050 0.00 0.50 /6 0.00 0.50 0.300 /3 0.350 0.400 0.450 0.500 0.550 0.600 0.650 /3 0.700 0.750 0.800 5/6 0.850 0.900 0.950 5 0 0.4633 0.059 0.0874 0.0649 0.035 0.034 0.0047 0.003 0.006 0.0005 0.000 0.0000 0.0000 0.890 0.5490 0.386 0.596 0.67 0.080 0.0353 0.094 0.04 0.005 0.007 0.0005 0.000 0.0000 0.0000 0.9638 0.859 0.604 0.53 0.3980 0.36 0.68 0.0794 0.067 0.07 0.007 0.0037 0.00 0.0003 0.000 0.0000 0.0000 3 0.9945 0.9444 0.87 0.7685 0.648 0.463 0.969 0.09 0.77 0.0905 0.044 0.076 0.0063 0.009 0.0005 0.0003 0.000 0.0000 4 0.9994 0.9873 0.9383 0.90 0.8358 0.6865 0.555 0.404 0.359 0.73 0.04 0.059 0.055 0.0093 0.008 0.008 0.0007 0.000 0.0000 5 0.9999 0.9978 0.983 0.976 0.9389 0.856 0.76 0.684 0.5643 0.403 0.608 0.509 0.0769 0.0338 0.04 0.0085 0.0037 0.0008 0.000 0.0000 0.0000 6.0000 0.9997 0.9964 0.9934 0.989 0.9434 0.8689 0.7970 0.7548 0.6098 0.45 0.3036 0.88 0.0950 0.04 0.0308 0.05 0.004 0.0008 0.000 0.000 7.0000 0.9994 0.9987 0.9958 0.987 0.9500 0.98 0.8868 0.7869 0.6535 0.5000 0.3465 0.3 0.3 0.088 0.0500 0.073 0.004 0.003 0.0006 0.0000 8 0.9999 0.9998 0.999 0.9958 0.9848 0.969 0.9578 0.9050 0.88 0.6964 0.5478 0.390 0.45 0.030 0.3 0.0566 0.08 0.0066 0.0036 0.0003 0.0000 9.0000.0000 0.9999 0.999 0.9963 0.995 0.9876 0.966 0.93 0.849 0.739 0.5968 0.4357 0.386 0.784 0.484 0.06 0.074 0.068 0.00 0.000 0.0000 0.9999 0.9993 0.998 0.997 0.9907 0.9745 0.9408 0.8796 0.787 0.648 0.5959 0.4845 0.335 0.64 0.0898 0.067 0.07 0.0006.0000 0.9999 0.9997 0.9995 0.998 0.9937 0.984 0.9576 0.9095 0.873 0.7908 0.703 0.5387 0.358 0.35 0.773 0.0556 0.0055.0000.0000 0.9999 0.9997 0.9989 0.9963 0.9893 0.979 0.9383 0.906 0.873 0.7639 0.600 0.4678 0.3958 0.84 0.036 3.0000.0000 0.9999 0.9995 0.9983 0.9948 0.9858 0.9806 0.9647 0.998 0.839 0.7404 0.684 0.450 0.70 4.0000.0000 0.9999 0.9995 0.9984 0.9977 0.9953 0.9866 0.9648 0.935 0.96 0.794 0.5367 5.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 6 0 0.440 0.853 0.0743 0.054 0.08 0.000 0.0033 0.005 0.000 0.0003 0.000 0.0000 0.0000 0.808 0.547 0.839 0.7 0.407 0.0635 0.06 0.037 0.0098 0.0033 0.000 0.0003 0.000 0.0000 0.957 0.789 0.564 0.4868 0.358 0.97 0.0994 0.0594 0.045 0.083 0.0066 0.00 0.0006 0.000 0.0000 0.0000 3 0.9930 0.936 0.7899 0.79 0.598 0.4050 0.459 0.659 0.339 0.065 0.08 0.006 0.0035 0.0009 0.000 0.000 0.0000 4 0.999 0.9830 0.909 0.8866 0.798 0.630 0.4499 0.339 0.89 0.666 0.0853 0.0384 0.049 0.0049 0.003 0.0008 0.0003 0.0000 5 0.9999 0.9967 0.9765 0.96 0.983 0.803 0.6598 0.5469 0.4900 0.388 0.976 0.05 0.0486 0.09 0.006 0.0040 0.006 0.0003 0.0000 6.0000 0.9995 0.9944 0.9899 0.9733 0.904 0.847 0.7374 0.688 0.57 0.3660 0.7 0.4 0.0583 0.09 0.059 0.007 0.006 0.000 0.0000 0.0000 7 0.9999 0.9989 0.9979 0.9930 0.979 0.956 0.8735 0.8406 0.76 0.569 0.408 0.559 0.43 0.067 0.0500 0.057 0.0075 0.005 0.0004 0.000 0.0000 8.0000 0.9998 0.9996 0.9985 0.995 0.9743 0.9500 0.939 0.8577 0.744 0.598 0.437 0.839 0.594 0.65 0.0744 0.07 0.0070 0.00 0.00 0.000 9.0000.0000 0.9998 0.9984 0.999 0.984 0.977 0.947 0.8759 0.778 0.6340 0.478 0.39 0.66 0.753 0.0796 0.067 0.00 0.0056 0.0005 0.0000 0.0000 0.9997 0.9984 0.9960 0.9938 0.9809 0.954 0.8949 0.804 0.67 0.500 0.453 0.340 0.897 0.087 0.0378 0.035 0.0033 0.000.0000 0.9997 0.999 0.9987 0.995 0.985 0.966 0.947 0.8334 0.708 0.6609 0.550 0.3698 0.08 0.34 0.079 0.070 0.0009.0000 0.9999 0.9998 0.999 0.9965 0.9894 0.979 0.9349 0.866 0.834 0.754 0.5950 0.409 0.709 0.0 0.0684 0.0070 3.0000.0000 0.9999 0.9994 0.9979 0.9934 0.987 0.9549 0.9406 0.9006 0.809 0.648 0.53 0.4386 0.08 0.049 4.0000 0.9999 0.9997 0.9990 0.9967 0.990 0.9863 0.9739 0.9365 0.8593 0.778 0.76 0.4853 0.89 5.0000.0000 0.9999 0.9997 0.9990 0.9985 0.9967 0.9900 0.979 0.9459 0.957 0.847 0.5599 6.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 7 0 0.48 0.668 0.063 0.045 0.05 0.0075 0.003 0.000 0.0007 0.000 0.0000 0.0000 0.79 0.488 0.55 0.983 0.8 0.050 0.093 0.0096 0.0067 0.00 0.0006 0.000 0.0000 0.0000 0.9497 0.768 0.598 0.4435 0.3096 0.637 0.0774 0.044 0.037 0.03 0.004 0.00 0.0003 0.000 0.0000 3 0.99 0.974 0.7556 0.6887 0.5489 0.3530 0.09 0.304 0.08 0.0464 0.084 0.0064 0.009 0.0005 0.000 0.0000 0.0000 4 0.9988 0.9779 0.903 0.8604 0.758 0.5739 0.3887 0.84 0.348 0.60 0.0596 0.045 0.0086 0.005 0.0006 0.0003 0.000 0.0000 5 0.9999 0.9953 0.968 0.9496 0.8943 0.7653 0.5968 0.4777 0.497 0.639 0.47 0.077 0.030 0.006 0.0030 0.009 0.0007 0.000 0.0000 6.0000 0.999 0.997 0.9853 0.963 0.899 0.775 0.6739 0.688 0.4478 0.90 0.66 0.086 0.0348 0.00 0.0080 0.003 0.0006 0.000 0.0000 7 0.9999 0.9983 0.9965 0.989 0.9598 0.8954 0.88 0.787 0.6405 0.4743 0.345 0.834 0.099 0.0383 0.073 0.07 0.003 0.0005 0.000 0.0000 8.0000 0.9997 0.9993 0.9974 0.9876 0.9597 0.945 0.9006 0.80 0.666 0.5000 0.3374 0.989 0.0994 0.0755 0.0403 0.04 0.006 0.0007 0.0003 0.0000 9.0000 0.9999 0.9995 0.9969 0.9873 0.977 0.967 0.908 0.866 0.6855 0.557 0.3595 0.8 0.79 0.046 0.040 0.009 0.0035 0.007 0.000 0.0000 0.9999 0.9994 0.9968 0.990 0.9880 0.965 0.974 0.8338 0.7098 0.55 0.38 0.36 0.48 0.07 0.0377 0.047 0.0083 0.0008 0.0000.0000 0.9999 0.9993 0.998 0.9970 0.9894 0.9699 0.983 0.859 0.736 0.5803 0.53 0.403 0.347 0.057 0.0504 0.039 0.0047 0.000.0000 0.9999 0.9997 0.9994 0.9975 0.994 0.9755 0.9404 0.8740 0.765 0.786 0.63 0.46 0.48 0.396 0.0987 0.0 0.00 3.0000.0000 0.9999 0.9995 0.998 0.9936 0.986 0.9536 0.897 0.8696 0.798 0.6470 0.45 0.33 0.444 0.086 0.0088 4.0000 0.9999 0.9997 0.9988 0.9959 0.9877 0.9673 0.9558 0.96 0.8363 0.6904 0.5565 0.480 0.38 0.0503 5.0000.0000 0.9999 0.9994 0.9979 0.9933 0.9904 0.9807 0.9499 0.888 0.807 0.7475 0.58 0.078 6.0000.0000 0.9998 0.9993 0.9990 0.9977 0.995 0.9775 0.9549 0.9369 0.833 0.589 7.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 5

CUMULATIVE BINOMIAL PROBABILITY x p 0.050 0.00 0.50 /6 0.00 0.50 0.300 /3 0.350 0.400 0.450 0.500 0.550 0.600 0.650 /3 0.700 0.750 0.800 5/6 0.850 0.900 0.950 8 0 0.397 0.50 0.0536 0.0376 0.080 0.0056 0.006 0.0007 0.0004 0.000 0.0000 0.0000 0.7735 0.4503 0.4 0.78 0.099 0.0395 0.04 0.0068 0.0046 0.003 0.0003 0.000 0.0000 0.949 0.7338 0.4794 0.407 0.73 0.353 0.0600 0.036 0.036 0.008 0.005 0.0007 0.000 0.0000 3 0.989 0.908 0.70 0.6479 0.500 0.3057 0.646 0.07 0.0783 0.038 0.00 0.0038 0.000 0.000 0.0000 0.0000 4 0.9985 0.978 0.8794 0.838 0.764 0.587 0.337 0.3 0.886 0.094 0.04 0.054 0.0049 0.003 0.0003 0.000 0.0000 5 0.9998 0.9936 0.958 0.9347 0.867 0.775 0.5344 0.4 0.3550 0.088 0.077 0.048 0.083 0.0058 0.004 0.0009 0.0003 0.0000 6.0000 0.9988 0.988 0.9794 0.9487 0.860 0.77 0.6085 0.549 0.3743 0.58 0.89 0.0537 0.003 0.006 0.0039 0.004 0.000 0.0000 7 0.9998 0.9973 0.9947 0.9837 0.943 0.8593 0.7767 0.783 0.5634 0.395 0.403 0.80 0.0576 0.0 0.044 0.006 0.00 0.000 0.0000 0.0000 8.0000 0.9995 0.9989 0.9957 0.9807 0.9404 0.894 0.8609 0.7368 0.5778 0.4073 0.57 0.347 0.0597 0.0433 0.00 0.0054 0.0009 0.000 0.000 9 0.9999 0.9998 0.999 0.9946 0.9790 0.9567 0.9403 0.8653 0.7473 0.597 0.4 0.63 0.39 0.076 0.0596 0.093 0.0043 0.00 0.0005 0.0000 0.0000.0000 0.9998 0.9988 0.9939 0.9856 0.9788 0.944 0.870 0.7597 0.6085 0.4366 0.77 0.33 0.407 0.0569 0.063 0.0053 0.007 0.000.0000 0.9998 0.9986 0.996 0.9938 0.9797 0.9463 0.88 0.774 0.657 0.4509 0.395 0.783 0.390 0.053 0.006 0.08 0.00 0.0000.0000 0.9997 0.999 0.9986 0.994 0.987 0.959 0.893 0.79 0.6450 0.5878 0.4656 0.85 0.39 0.0653 0.049 0.0064 0.000 3.0000 0.9999 0.9997 0.9987 0.995 0.9846 0.9589 0.9058 0.84 0.7689 0.6673 0.483 0.836 0.68 0.06 0.08 0.005 4.0000.0000 0.9998 0.9990 0.996 0.9880 0.967 0.97 0.8983 0.8354 0.6943 0.4990 0.35 0.798 0.098 0.009 5.0000 0.9999 0.9993 0.9975 0.998 0.9764 0.9674 0.9400 0.8647 0.787 0.5973 0.503 0.66 0.058 6.0000 0.9999 0.9997 0.9987 0.9954 0.993 0.9858 0.9605 0.9009 0.87 0.7759 0.5497 0.65 7.0000.0000 0.9999 0.9996 0.9993 0.9984 0.9944 0.980 0.964 0.9464 0.8499 0.608 8.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 9 0 0.3774 0.35 0.0456 0.033 0.044 0.004 0.00 0.0005 0.0003 0.000 0.0000 0.7547 0.403 0.985 0.50 0.089 0.030 0.004 0.0047 0.003 0.0008 0.000 0.0000 0.0000 0.9335 0.7054 0.443 0.3643 0.369 0.3 0.046 0.040 0.070 0.0055 0.005 0.0004 0.000 0.0000 3 0.9868 0.8850 0.684 0.6070 0.455 0.63 0.33 0.0787 0.059 0.030 0.0077 0.00 0.0005 0.000 0.0000 0.0000 4 0.9980 0.9648 0.8556 0.80 0.6733 0.4654 0.8 0.879 0.500 0.0696 0.080 0.0096 0.008 0.0006 0.000 0.000 0.0000 5 0.9998 0.994 0.9463 0.976 0.8369 0.6678 0.4739 0.359 0.968 0.69 0.0777 0.038 0.009 0.003 0.0007 0.0004 0.000 0.0000 6.0000 0.9983 0.9837 0.979 0.934 0.85 0.6655 0.543 0.48 0.308 0.77 0.0835 0.034 0.06 0.003 0.009 0.0006 0.000 7 0.9997 0.9959 0.99 0.9767 0.95 0.880 0.707 0.6656 0.4878 0.369 0.796 0.087 0.035 0.04 0.0074 0.008 0.0005 0.0000 0.0000 8.0000 0.999 0.998 0.9933 0.973 0.96 0.8538 0.845 0.6675 0.4940 0.338 0.84 0.0885 0.0347 0.04 0.005 0.003 0.0003 0.000 0.0000 9 0.9999 0.9996 0.9984 0.99 0.9674 0.935 0.95 0.839 0.670 0.5000 0.390 0.86 0.0875 0.0648 0.036 0.0089 0.006 0.0004 0.000 0.0000 0.9999 0.9997 0.9977 0.9895 0.9759 0.9653 0.95 0.859 0.676 0.5060 0.335 0.855 0.46 0.0839 0.087 0.0067 0.008 0.0008 0.0000.0000.0000 0.9995 0.997 0.996 0.9886 0.9648 0.99 0.804 0.683 0.5 0.3344 0.793 0.80 0.0775 0.033 0.0079 0.004 0.0003 0.9999 0.9994 0.998 0.9969 0.9884 0.9658 0.965 0.873 0.699 0.588 0.4569 0.3345 0.749 0.0676 0.08 0.063 0.007 3.0000 0.9999 0.9996 0.9993 0.9969 0.989 0.968 0.93 0.837 0.703 0.648 0.56 0.33 0.63 0.084 0.0537 0.0086 0.000 4.0000 0.9999 0.9999 0.9994 0.997 0.9904 0.970 0.9304 0.8500 0.8 0.778 0.5346 0.367 0.989 0.444 0.035 0.000 5.0000.0000 0.9999 0.9995 0.9978 0.993 0.9770 0.9409 0.93 0.8668 0.7369 0.5449 0.3930 0.359 0.50 0.03 6.0000 0.9999 0.9996 0.9985 0.9945 0.9830 0.9760 0.9538 0.8887 0.763 0.6357 0.5587 0.946 0.0665 7.0000.0000 0.9998 0.999 0.9969 0.9953 0.9896 0.9690 0.97 0.8498 0.805 0.5797 0.453 8.0000 0.9999 0.9997 0.9995 0.9989 0.9958 0.9856 0.9687 0.9544 0.8649 0.66 9.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000 6

CUMULATIVE BINOMIAL PROBABILITY x p 0.050 0.00 0.50 /6 0.00 0.50 0.300 /3 0.350 0.400 0.450 0.500 0.550 0.600 0.650 /3 0.700 0.750 0.800 5/6 0.850 0.900 0.950 0 0 0.3585 0.6 0.0388 0.06 0.05 0.003 0.0008 0.0003 0.000 0.0000 0.0000 0.7358 0.397 0.756 0.304 0.069 0.043 0.0076 0.0033 0.00 0.0005 0.000 0.0000 0.945 0.6769 0.4049 0.387 0.06 0.093 0.0355 0.076 0.0 0.0036 0.0009 0.000 0.0000 3 0.984 0.8670 0.6477 0.5665 0.44 0.5 0.07 0.0604 0.0444 0.060 0.0049 0.003 0.0003 0.0000 4 0.9974 0.9568 0.898 0.7687 0.696 0.448 0.375 0.55 0.8 0.050 0.089 0.0059 0.005 0.0003 0.0000 0.0000 5 0.9997 0.9887 0.937 0.898 0.804 0.67 0.464 0.97 0.454 0.56 0.0553 0.007 0.0064 0.006 0.0003 0.000 0.0000 6.0000 0.9976 0.978 0.969 0.933 0.7858 0.6080 0.4793 0.466 0.500 0.99 0.0577 0.04 0.0065 0.005 0.0009 0.0003 0.0000 7 0.9996 0.994 0.9887 0.9679 0.898 0.773 0.665 0.600 0.459 0.50 0.36 0.0580 0.00 0.0060 0.0037 0.003 0.000 0.0000 8 0.9999 0.9987 0.997 0.9900 0.959 0.8867 0.8095 0.764 0.5956 0.443 0.57 0.308 0.0565 0.096 0.030 0.005 0.0009 0.000 0.0000 9.0000 0.9998 0.9994 0.9974 0.986 0.950 0.908 0.878 0.7553 0.594 0.49 0.493 0.75 0.053 0.0376 0.07 0.0039 0.0006 0.000 0.0000 0.0000 0.9999 0.9994 0.996 0.989 0.964 0.9468 0.875 0.7507 0.588 0.4086 0.447 0.8 0.099 0.0480 0.039 0.006 0.0006 0.000 0.0000.0000 0.9999 0.999 0.9949 0.9870 0.9804 0.9435 0.869 0.7483 0.5857 0.4044 0.376 0.905 0.33 0.0409 0.000 0.008 0.003 0.000.0000 0.9998 0.9987 0.9963 0.9940 0.9790 0.940 0.8684 0.7480 0.584 0.3990 0.3385 0.77 0.08 0.03 0.03 0.0059 0.0004 3.0000 0.9997 0.999 0.9985 0.9935 0.9786 0.943 0.870 0.7500 0.5834 0.507 0.390 0.4 0.0867 0.037 0.09 0.004 0.0000 4.0000 0.9998 0.9998 0.9984 0.9936 0.9793 0.9447 0.8744 0.7546 0.708 0.5836 0.388 0.958 0.08 0.0673 0.03 0.0003 5.0000.0000 0.9997 0.9985 0.994 0.98 0.9490 0.888 0.8485 0.765 0.585 0.3704 0.33 0.70 0.043 0.006 6.0000 0.9997 0.9987 0.995 0.9840 0.9556 0.9396 0.899 0.7748 0.5886 0.4335 0.353 0.330 0.059 7.0000 0.9998 0.999 0.9964 0.9879 0.984 0.9645 0.9087 0.7939 0.673 0.595 0.33 0.0755 8.0000 0.9999 0.9995 0.9979 0.9967 0.994 0.9757 0.9308 0.8696 0.844 0.6083 0.64 9.0000.0000 0.9998 0.9997 0.999 0.9968 0.9885 0.9739 0.96 0.8784 0.645 0.0000.0000.0000.0000.0000.0000.0000.0000.0000 7

CUMULATIVE POISSON PROBABILITY The Poisso distributio: cumulative probabilities P(X x) = r=0 e x λ λr r! x λ 0.0 0.0 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 0.9900 0.980 0.9704 0.9608 0.95 0.948 0.934 0.93 0.939.0000 0.9998 0.9996 0.999 0.9988 0.9983 0.9977 0.9970 0.996......0000.0000.0000.0000.0000 0.9999 0.9999 0.9999 3...............................0000.0000.0000 x λ 0.0 0.0 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0 0.9048 0.887 0.7408 0.6703 0.6065 0.5488 0.4966 0.4493 0.4066 0.9953 0.985 0.963 0.9384 0.9098 0.878 0.844 0.8088 0.775 0.9998 0.9989 0.9964 0.99 0.9856 0.9769 0.9659 0.956 0.937 3.0000 0.9999 0.9997 0.999 0.998 0.9966 0.994 0.9909 0.9865 4......0000.0000 0.9999 0.9998 0.9996 0.999 0.9986 0.9977 5................0000.0000.0000 0.9999 0.9998 0.9997 6...............................0000.0000.0000 7............................................. x λ 3.00 3.0 3.0 3.30 3.40 3.50 3.60 3.70 3.80 3.90 0 0.0498 0.0450 0.0408 0.0369 0.0334 0.030 0.073 0.047 0.04 0.00 0.99 0.847 0.7 0.586 0.468 0.359 0.57 0.6 0.074 0.099 0.43 0.40 0.3799 0.3594 0.3397 0.308 0.307 0.854 0.689 0.53 3 0.647 0.648 0.605 0.5803 0.5584 0.5366 0.55 0.494 0.4735 0.453 4 0.853 0.798 0.7806 0.766 0.744 0.754 0.7064 0.687 0.6678 0.6484 5 0.96 0.9057 0.8946 0.889 0.8705 0.8576 0.844 0.830 0.856 0.8006 6 0.9665 0.96 0.9554 0.9490 0.94 0.9347 0.967 0.98 0.909 0.8995 7 0.988 0.9858 0.983 0.980 0.9769 0.9733 0.969 0.9648 0.9599 0.9546 8 0.996 0.9953 0.9943 0.993 0.997 0.990 0.9883 0.9863 0.9840 0.985 9 0.9989 0.9986 0.998 0.9978 0.9973 0.9967 0.9960 0.995 0.994 0.993 0 0.9997 0.9996 0.9995 0.9994 0.999 0.9990 0.9987 0.9984 0.998 0.9977 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997 0.9996 0.9995 0.9994 0.9993.0000.0000.0000.0000 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 3.....................0000.0000.0000.0000.0000 0.9999 4..............................................0000 x λ.00.0.0.30.40.50.60.70.80.90 0 0.3679 0.339 0.30 0.75 0.466 0.3 0.09 0.87 0.653 0.496 0.7358 0.6990 0.666 0.668 0.598 0.5578 0.549 0.493 0.468 0.4337 0.997 0.9004 0.8795 0.857 0.8335 0.8088 0.7834 0.757 0.7306 0.7037 3 0.980 0.9743 0.966 0.9569 0.9463 0.9344 0.9 0.9068 0.893 0.8747 4 0.9963 0.9946 0.993 0.9893 0.9857 0.984 0.9763 0.9704 0.9636 0.9559 5 0.9994 0.9990 0.9985 0.9978 0.9968 0.9955 0.9940 0.990 0.9896 0.9868 6 0.9999 0.9999 0.9997 0.9996 0.9994 0.999 0.9987 0.998 0.9974 0.9966 7.0000.0000.0000 0.9999 0.9999 0.9998 0.9997 0.9996 0.9994 0.999 8................0000.0000.0000.0000 0.9999 0.9999 0.9998 9....................................0000.0000.0000 x λ.00.0.0.30.40.50.60.70.80.90 0 0.353 0.5 0.08 0.003 0.0907 0.08 0.0743 0.067 0.0608 0.0550 0.4060 0.3796 0.3546 0.3309 0.3084 0.873 0.674 0.487 0.3 0.46 0.6767 0.6496 0.67 0.5960 0.5697 0.5438 0.584 0.4936 0.4695 0.4460 3 0.857 0.8386 0.894 0.7993 0.7787 0.7576 0.7360 0.74 0.699 0.6696 4 0.9473 0.9379 0.975 0.96 0.904 0.89 0.8774 0.869 0.8477 0.838 5 0.9834 0.9796 0.975 0.9700 0.9643 0.9580 0.950 0.9433 0.9349 0.958 6 0.9955 0.994 0.995 0.9906 0.9884 0.9858 0.988 0.9794 0.9756 0.973 7 0.9989 0.9985 0.9980 0.9974 0.9967 0.9958 0.9947 0.9934 0.999 0.990 8 0.9998 0.9997 0.9995 0.9994 0.999 0.9989 0.9985 0.998 0.9976 0.9969 9.0000 0.9999 0.9999 0.9999 0.9998 0.9997 0.9996 0.9995 0.9993 0.999 0......0000.0000.0000.0000 0.9999 0.9999 0.9999 0.9998 0.9998..........................0000.0000.0000.0000 0.9999..............................................0000 x λ 4.00 4.0 4.0 4.30 4.40 4.50 4.60 4.70 4.80 4.90 0 0.083 0.066 0.050 0.036 0.03 0.0 0.00 0.009 0.008 0.0074 0.096 0.0845 0.0780 0.079 0.0663 0.06 0.0563 0.058 0.0477 0.0439 0.38 0.38 0.0 0.974 0.85 0.736 0.66 0.53 0.45 0.333 3 0.4335 0.44 0.3954 0.377 0.3594 0.343 0.357 0.3097 0.94 0.793 4 0.688 0.6093 0.5898 0.5704 0.55 0.53 0.53 0.4946 0.4763 0.458 5 0.785 0.7693 0.753 0.7367 0.799 0.709 0.6858 0.6684 0.650 0.6335 6 0.8893 0.8786 0.8675 0.8558 0.8436 0.83 0.880 0.8046 0.7908 0.7767 7 0.9489 0.947 0.936 0.990 0.94 0.934 0.9049 0.8960 0.8867 0.8769 8 0.9786 0.9755 0.97 0.9683 0.964 0.9597 0.9549 0.9497 0.944 0.938 9 0.999 0.9905 0.9889 0.987 0.985 0.989 0.9805 0.9778 0.9749 0.977 0 0.997 0.9966 0.9959 0.995 0.9943 0.9933 0.99 0.990 0.9896 0.9880 0.999 0.9989 0.9986 0.9983 0.9980 0.9976 0.997 0.9966 0.9960 0.9953 0.9997 0.9997 0.9996 0.9995 0.9993 0.999 0.9990 0.9988 0.9986 0.9983 3 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997 0.9997 0.9996 0.9995 0.9994 4.0000.0000.0000.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 5.....................0000.0000.0000.0000.0000 0.9999 6..............................................0000 8

CUMULATIVE POISSON PROBABILITY x λ 5.00 5.0 5.0 5.30 5.40 5.50 5.60 5.70 5.80 5.90 0 0.0067 0.006 0.0055 0.0050 0.0045 0.004 0.0037 0.0033 0.0030 0.007 0.0404 0.037 0.034 0.034 0.089 0.066 0.044 0.04 0.006 0.089 0.47 0.65 0.088 0.06 0.0948 0.0884 0.084 0.0768 0.075 0.0666 3 0.650 0.53 0.38 0.54 0.33 0.07 0.906 0.800 0.700 0.604 4 0.4405 0.43 0.406 0.3895 0.3733 0.3575 0.34 0.37 0.37 0.987 5 0.660 0.5984 0.5809 0.5635 0.546 0.589 0.59 0.4950 0.4783 0.469 6 0.76 0.7474 0.734 0.77 0.707 0.6860 0.6703 0.6544 0.6384 0.64 7 0.8666 0.8560 0.8449 0.8335 0.87 0.8095 0.7970 0.784 0.770 0.7576 8 0.939 0.95 0.98 0.906 0.907 0.8944 0.8857 0.8766 0.867 0.8574 9 0.968 0.9644 0.9603 0.9559 0.95 0.946 0.9409 0.935 0.99 0.98 0 0.9863 0.9844 0.983 0.9800 0.9775 0.9747 0.978 0.9686 0.965 0.964 0.9945 0.9937 0.997 0.996 0.9904 0.9890 0.9875 0.9859 0.984 0.98 0.9980 0.9976 0.997 0.9967 0.996 0.9955 0.9949 0.994 0.993 0.99 3 0.9993 0.999 0.9990 0.9988 0.9986 0.9983 0.9980 0.9977 0.9973 0.9969 4 0.9998 0.9997 0.9997 0.9996 0.9995 0.9994 0.9993 0.999 0.9990 0.9988 5 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 0.9998 0.9997 0.9996 0.9996 6.0000.0000.0000.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 7.....................0000.0000.0000.0000.0000.0000 x λ 6.00 6.0 6.0 6.30 6.40 6.50 6.60 6.70 6.80 6.90 0 0.005 0.00 0.000 0.008 0.007 0.005 0.004 0.00 0.00 0.000 0.074 0.059 0.046 0.034 0.03 0.03 0.003 0.0095 0.0087 0.0080 0.060 0.0577 0.0536 0.0498 0.0463 0.0430 0.0400 0.037 0.0344 0.030 3 0.5 0.45 0.34 0.64 0.89 0.8 0.05 0.0988 0.098 0.087 4 0.85 0.79 0.59 0.469 0.35 0.37 0.7 0.0 0.90 0.83 5 0.4457 0.498 0.44 0.3988 0.3837 0.3690 0.3547 0.3406 0.370 0.337 6 0.6063 0.590 0.574 0.558 0.543 0.565 0.508 0.4953 0.4799 0.4647 7 0.7440 0.730 0.760 0.707 0.6873 0.678 0.658 0.6433 0.685 0.636 8 0.847 0.8367 0.859 0.848 0.8033 0.796 0.7796 0.7673 0.7548 0.740 9 0.96 0.9090 0.906 0.8939 0.8858 0.8774 0.8686 0.8596 0.850 0.8405 0 0.9574 0.953 0.9486 0.9437 0.9386 0.933 0.974 0.94 0.95 0.9084 0.9799 0.9776 0.9750 0.973 0.9693 0.966 0.967 0.959 0.955 0.950 0.99 0.9900 0.9887 0.9873 0.9857 0.9840 0.98 0.980 0.9779 0.9755 3 0.9964 0.9958 0.995 0.9945 0.9937 0.999 0.990 0.9909 0.9898 0.9885 4 0.9986 0.9984 0.998 0.9978 0.9974 0.9970 0.9966 0.996 0.9956 0.9950 5 0.9995 0.9994 0.9993 0.999 0.9990 0.9988 0.9986 0.9984 0.998 0.9979 6 0.9998 0.9998 0.9997 0.9997 0.9996 0.9996 0.9995 0.9994 0.9993 0.999 7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 0.9998 0.9997 0.9997 8.0000.0000.0000.0000.0000 0.9999 0.9999 0.9999 0.9999 0.9999 9..........................0000.0000.0000.0000.0000 x λ 7.00 7.0 7.0 7.30 7.40 7.50 7.60 7.70 7.80 7.90 0 0.0009 0.0008 0.0007 0.0007 00006 0.0006 0.0005 0.0005 0.0004 0.0004 0.0073 0.0067 0.006 0.0056 0.005 0.0047 0.0043 0.0039 0.0036 0.0033 0.096 0.075 0.055 0.036 0.09 0.003 0.088 0.074 0.06 0.049 3 0.088 0.0767 0.079 0.0674 0.063 0.059 0.0554 0.058 0.0485 0.0453 4 0.730 0.64 0.555 0.473 0.395 0.3 0.49 0.8 0.7 0.055 5 0.3007 0.88 0.759 0.640 0.56 0.44 0.307 0.03 0.03 0.006 6 0.4497 0.4349 0.404 0.4060 0.390 0.378 0.3646 0.354 0.3384 0.357 7 0.5987 0.5838 0.5689 0.554 0.5393 0.546 0.500 0.4956 0.48 0.4670 8 0.79 0.760 0.707 0.689 0.6757 0.660 0.648 0.6343 0.604 0.6065 9 0.8305 0.80 0.8096 0.7988 0.7877 0.7764 0.7649 0.753 0.74 0.790 0 0.905 0.894 0.8867 0.8788 0.8707 0.86 0.8535 0.8445 0.835 0.857 0.9467 0.940 0.937 0.939 0.965 0.908 0.948 0.9085 0.900 0.895 0.9730 0.9703 0.9673 0.964 0.9609 0.9573 0.9536 0.9496 0.9454 0.9409 3 0.987 0.9857 0.984 0.984 0.9805 0.9784 0.976 0.9739 0.974 0.9687 4 0.9943 0.9935 0.997 0.998 0.9908 0.9897 0.9886 0.9873 0.9859 0.9844 5 0.9976 0.997 0.9969 0.9964 0.9959 0.9954 0.9948 0.994 0.9934 0.996 6 0.9990 0.9989 0.9987 0.9985 0.9983 0.9980 09978 0.9974 0.997 0.9967 7 0.9996 0.9996 0.9995 0.9994 0.9993 0.999 0.999 0.9989 0.9988 0.9986 8 0.9999 0.9998 0.9998 0.9998 0.9997 0.9997 0.9996 0.9996 0.9995 0.9994 9.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 0.9998 0......0000.0000.0000.0000.0000.0000 0.9999 0.9999 0.9999....................................0000.0000.0000 x λ 8.00 8.0 8.0 8.30 8.40 8.50 8.60 8.70 8.80 8.90 0 0.0003 0.0003 0.0003 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.0030 0.008 0.005 0.003 0.00 0.009 0.008 0.006 0.005 0.004 0.038 0.07 0.08 0.009 0.000 0.0093 0.0086 0.0079 0.0073 0.0068 3 0.044 0.0396 0.0370 0.0346 0.033 0.030 0.08 0.06 0.044 0.08 4 0.0996 0.0940 0.0887 0.0837 0.0789 0.0744 0.070 0.0660 0.06 0.0584 5 0.9 0.8 0.736 0.653 0.573 0.496 0.4 0.35 0.84 0.9 6 0.334 0.303 0.896 0.78 0.670 0.56 0.457 0.355 0.56 0.60 7 0.4530 0.439 0.454 0.49 0.3987 0.3856 0.378 0.360 0.3478 0.3357 8 0.595 0.5786 0.5647 0.5507 0.5369 0.53 0.5094 0.4958 0.483 0.4689 9 0.766 0.704 0.695 0.6788 0.6659 0.6530 0.6400 0.669 0.637 0.6006 0 0.859 0.8058 0.7955 0.7850 0.7743 0.7634 0.75 0.7409 0.794 0.778 0.888 0.8807 0.873 0.865 0.857 0.8487 0.8400 0.83 0.80 0.86 0.936 0.933 0.96 0.907 0.950 0.909 0.909 0.8965 0.8898 0.889 3 0.9658 0.968 0.9595 0.956 0.954 0.9486 0.9445 0.9403 0.9358 0.93 4 0.987 0.980 0.979 0.977 0.9749 0.976 0.970 0.9675 0.9647 0.967 5 0.998 0.9908 0.9898 0.9887 0.9875 0.986 0.9848 0.983 0.986 0.9798 6 0.9963 0.9958 0.9953 0.9947 0.994 0.9934 0.996 0.998 0.9909 0.9899 7 0.9984 0.998 0.9979 0.9977 0.9973 0.9970 0.9966 0.996 0.9957 0.995 8 0.9993 0.999 0.999 0.9990 0.9989 0.9987 0.9985 0.9983 0.998 0.9978 9 0.9997 0.9997 0.9997 0.9996 0.9995 0.9995 0.9994 0.9993 0.999 0.999 0 0.9999 0.9999 0.9999 0.9998 0.9998 0.9998 0.9998 0.9997 0.9997 0.9996.0000.0000.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998................0000.0000.0000.0000.0000.0000 0.9999 3..............................................0000 9