Pearson Edexcel Level 3 Advanced Subsidiary And Advanced GCE .

MPHStatisticsDiscrete distributionsFor a discrete random variableExpectation (mean):X taking values xi with probabilities P(X xi )E(X ) μ Σxi P(X xi )Var(X ) σ 2 Σ(xi – μ)2 P(X xi ) Σ x2i P(X xi ) – μ 2Variance:Discrete distributionsStandard discrete distributions:Distribution ofBinomialP(X x)MeanVariance n xn –x x p (1 – p)npnp(1 – p)λλXB(n, p)e λPo(λ)Poissonλxx!Continuous distributionsFor a continuous random variableExpectation (mean):Variance:3X having probability density function fE(X ) μ x f(x) dxVar(X ) σ 2 (x – μ)2 f(x) dx x2 f(x) dx – μ 2For a functiong(X ): E(g(X )) g(x) f(x) dxCumulative distribution function:F(x0) P(X x0 ) #x0f (t) dt Standard continuous distribution:Distribution ofNormalXP.D.F.1 x μ σ 1e 2 σ 2πN(μ, σ 2)Uniform (Rectangular) on[a, b]1b aMeanVarianceμσ2212(a b)112(b – a)2Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further MathematicsMathematical Formulae and Statistical Tables – Issue 1 – July 2017 – Pearson Education Limited 201713

MPHCorrelation and regressionFor a set ofn pairs of values (xi , yi )(Σ xi ) 2Sxx Σ(xi – x) Σx –n22i(Σ y i ) 2Syy Σ( yi – y) Σy –n22iSxy Σ(xi – x)( yi – y) Σxi yi –(Σ xi )(Σ y i )nThe product moment correlation coefficient isr SxySxxS yy {Σ( xi x )( y i y )Σ( xi x ) 2The regression coefficient of}{Σ( y i y ) 2y on x is b Least squares regression line ofResidual Sum of Squares (RSS)}SxySxx (Σ xi )(Σ y i )n2 2 (Σ xi ) 2 (Σ y i ) 2 Σ xi n Σ y i n Σ xi y i Σ( xi x )( y i y )Σ( xi x ) 2y on x is y a bx where a y – bx Syy(S )–Spearman’s rank correlation coefficient isxySxx2 Syy (1 – r 2)rS 1 –6Σd 2n(n 2 1)Non-parametric testsGoodness-of-fit test and contingency tables:(Oi Ei ) 2 E χ v2iStatistical tablesThe following statistical tables are required for AS Level Further Mathematics:Binomial Cumulative Distribution Function (see page 29)Poisson Cumulative Distribution Function (see page 35)Percentage Points of theχ2 Distribution (see page 36)Critical Values for Correlation Coefficients: Product Moment Coefficient and Spearman’s Coefficient(see page 37)Random Numbers (see page 38)14Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further MathematicsMathematical Formulae and Statistical Tables – Issue 1 – July 2017 – Pearson Education Limited 2017

MPHMechanicsCentres of massFor uniform bodies:Triangular lamina:23along median from vertexCircular arc, radius r, angle at centre2α :Sector of circle, radius r, angle at centrer sin αα2α :from centre2r sin α3αfrom centre3Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further MathematicsMathematical Formulae and Statistical Tables – Issue 1 – July 2017 – Pearson Education Limited 201715

MPH16Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further MathematicsMathematical Formulae and Statistical Tables – Issue 1 – July 2017 – Pearson Education Limited 2017

MPH4 A Level Further MathematicsStudents sitting an A Level Further Mathematics paper may also require those formulae listed forA Level Mathematics in Section 2.Pure MathematicsSummationsn r 6 n(n 1)(2n 1)3 4 n2(n 1)2r 1n r12r 11Matrix transformationsAnticlockwise rotation throughReflection in the line cos θθ about O: sin θ cos 2θy (tanθ)x : sin 2θ sin θ cos θ sin 2θ cos 2θ Area of a sectorA 12 r2dθ(polar coordinates)Complex numbers{r(cosθ isin θ)}n r n (cosnθ isin nθ)The roots ofz n 1 are given by z e2 πk in, fork 0, 1, 2, . . ., n – 14Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further MathematicsMathematical Formulae and Statistical Tables – Issue 1 – July 2017 – Pearson Education Limited 201717

MPHMaclaurin’s and Taylor’s Seriesx2x r (r)f (x) f(0) x f′(0) f′′(0) . . . f (0) . . .2!r!e x exp(x) 1 x x2xr . .2!r!for allx2x3xr – . . . (–1) r 1 .ln(1 x) x –23rsin x x –x 2 r 1x3x5 – . . . (–1) r .(2r 1)!3!5!cos x 1 –x2rx2x4 – . . . (–1) r .(2r )!2!4!(–1 x 1)for allfor allx3x5x 2 r 1r – . . . (–1) .arctan x x –352r 1VectorsiVector product: a b ½a½½b½sinθ n̂ a1b1a1a.(b c) b1c1a2b2c2xja2b2xx(–1 x 1)k a2b3 a3b2 a3 a3b1 a1b3 b3 a1b2 a2b1 a3b3 b.(c a) c.(a b)c3A is the point with position vector a a1i a2 j a3k and the direction vector b is given byb b1i b2 j b3k, then the straight line through A with direction vector b has cartesian equationIfz a3x a1y a2 ( λ)b1b2b3The plane throughA with normal vector n n1i n2 j n3k has cartesian equationn1 x n2 y n3 z d 0 where d –a.nThe plane through non-collinear pointsA, B and C has vector equationr a λ(b – a) μ(c – a) (1 – λ – μ)a λb μcThe plane through the point with position vectora and parallel to b and c has equationr a sb tcThe perpendicular distance of18(α, β, γ)fromn1 x n2 y n3 z d 0 isn1α n2 β n3 γ dn12 n22 n32.Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further MathematicsMathematical Formulae and Statistical Tables – Issue 1 – July 2017 – Pearson Education Limited 2017

MPHHyperbolic functionscosh2 x – sinh2 x 1sinh 2x 2sinh x cosh xcosh 2x cosh2 x sinh2 xarcosh x ln{x x 2 1}arsinh x ln{x x 2 1}artanh x 12 1 ln 1 x x (x 1)(½x½ lax2 y 2 1a 2 b2y 4axx2y2 1a 2 b2xy c 2Parametric Form(a cosθ, b sinθ)(at 2, 2at)(a sec θ, b tan θ)( a coshθ, b sinh θ)c ct , tEccentricitye 1b 2 a 2 (1 – e 2)e 1e 1b 2 a 2 (e 2 – 1)Foci( ae, 0)(a, 0)( ae, 0)Directricesx x –ax Standard FormAsymptotesaenone2noneaee ( 2 c, 2 c)x y 2cxy abPearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further MathematicsMathematical Formulae and Statistical Tables – Issue 1 – July 2017 – Pearson Education Limited 20172x 0, y 0194

MPHDifferentiationf (x)f ′ (x)1arcsin x1 x21arccos x–arctan x11 x2sinh xcosh xcosh xsinh xtanh xsech2 xarsinh xarcosh xartanh x201 x211 x21x2 111 x2Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further MathematicsMathematical Formulae and Statistical Tables – Issue 1 – July 2017 – Pearson Education Limited 2017

MPHIntegration ( constant;a 0 where relevant)f (x) f (x) dxsinh xcosh xcosh xsinh xtanh xlncosh x12a x x arcsin a 21 x arctan a a1a x2212x a212(½x½ a)a x2 x arcosh , ln{x a x 2 a 2 } (x a) x arsinh , ln{x a x2 a2 }1a x211a x x ln artanh a 2aaa x1x a21x aln2ax a22(½x½ a)Arc lengths # dy 1 dx dx s # dx dy dtdtdts # dr r 2 dθ dθ 22(cartesian coordinates)2(parametric form)42(polar form)Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further MathematicsMathematical Formulae and Statistical Tables – Issue 1 – July 2017 – Pearson Education Limited 201721

MPHSurface area of revolutions x 2π#2 dy y 1 d x dx sx 2π# dx dy y dt dt dt #22(parametric form)2 dr sx 2π r sin θ r d θ dθ 22(cartesian coordinates)2(polar form)Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further MathematicsMathematical Formulae and Statistical Tables – Issue 1 – July 2017 – Pearson Education Limited 2017

MPHStatisticsDiscrete distributionsFor a discrete random variableExpectation (mean):X taking values xi with probabilities P(X xi )E(X ) μ Σxi P(X xi )Var(X ) σ 2 Σ(xi – μ)2 P(X xi ) Σ x2i P(X xi ) – μ 2Variance:For a functiong(X ): E(g(X )) Σg(xi ) P(X xi )The probability generating function ofX is GX (t) E(t X ) andE(X ) G′X (1) and Var(X ) G′′X (1) G′X (1) – [G′X (1)]2ForZ X Y, where X and Y are independent: GZ (t) GX (t) GY (t)Discrete distributionsStandard discrete distributions:Distribution ofBinomialPoissonB(n, p)Po(λ)GeometriconX1, 2, . . .Geo( p)Negative binomialonr, r 1, . . .P(X x)MeanVarianceP.G.F. n xn –x x p (1 – p)npnp(1 – p)(1 – p pt)nλλeλ(t –1)p(1 – p)x –11p1 pp2pt1 (1 p )t x 1 rx –r r 1 p (1 – p)rpr (1 p )p2 pt 1 (1 p )t λxx!e λr4Continuous distributionsFor a continuous random variableExpectation (mean):Variance:X having probability density function fE(X ) μ x f(x) dxVar(X ) σ 2 (x – μ)2 f(x) dx x2 f(x) dx – μ 2For a functiong(X ): E(g(X )) g(x) f(x) dxCumulative distribution function:F(x0) P(X x0 ) #x0f (t) dt Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further MathematicsMathematical Formulae and Statistical Tables – Issue 1 – July 2017 – Pearson Education Limited 201723

MPHStandard continuous distribution:Distribution ofNormalXP.D.F.1 x μ σ 1e 2 σ 2πN(μ, σ 2)Uniform (Rectangular) onVarianceμσ221b a[a, b]Mean12(a b)112(b – a)2Correlation and regressionFor a set ofn pairs of values (xi , yi )Sxx Σ(xi – x)2 Σxi2 –(Σ xi ) 2nSyy Σ( yi – y)2 Σyi2 –(Σ y i ) 2nSxy Σ(xi – x)( yi – y) Σxi yi –(Σ xi )(Σ y i )nThe product moment correlation coefficient isr SxySxxS yy Σ( xi x )( y i y ){Σ( xi x )2The regression coefficient of}{Σ( yi y )2Sxyy on x is b Least squares regression line ofResidual Sum of Squares (RSS)}Sxx (Σ xi )(Σ y i )n2 2 (Σ xi ) 2 (Σ y i ) 2 Σ xi n Σ y i n Σ xi y i Σ( xi x )( y i y )Σ( xi x ) 2y on x is y a bx where a y – bx Syy(S )–xySxx2 Syy (1 – r 2)6Σd 2Spearman’s rank correlation coefficient is rs 1 –n(n 2 1)Expectation algebraFor independent random variablesX and YE(XY ) E(X )E(Y ), Var(aX bY ) a2 Var(X ) b2 Var(Y )24Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further MathematicsMathematical Formulae and Statistical Tables – Issue 1 – July 2017 – Pearson Education Limited 2017

MPHSampling distributions(i)σ is knownTests for mean whenX1, X2 , . . ., Xn of n independent observations from a distribution havingmean μ and variance σ 2:For a random sampleX is an unbiased estimator of μ, with Var(X ) S 2 is an unbiased estimator of σ 2, where S 2 For a random sample of For a random sample ofsample ofσ2nΣ( X i X ) 2n 1nx observations from N(μx, σ 2x ) and, independently, a randomny observations from N(μy , σ 2y ),( X Y ) ( μx μ y )σ 2xnx(ii)Tests for variance and mean whenFor a random sample ofX μ N(0, 1)σ/ nn observations from N(μ, σ 2), σ 2y N(0, 1)nyσ is not knownn observations from N( μ, σ 2 )(n 1) S 2 χ n2 –1σ2X μ tn –1S/ n(also valid in matched-pairs situations)nx observations from N(μx, σ 2x ) and, independently, a random sampleof ny observations from N(μy, σ 2y ) For a random sample ofS 2x / σ 2xS 2y / σ 2yIf Fn x – 1, ny – 14σ 2x σ 2y σ 2 (unknown) then( X Y ) ( μx μ y ) 11 S p2 nx n y t nx n y 2where2pS (n x 1) S 2x (n y 1) S 2ynx n y 2Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further MathematicsMathematical Formulae and Statistical Tables – Issue 1 – July 2017 – Pearson Education Limited 201725

MPHNon-parametric testsGoodness-of-fit test and contingency tables:(Oi Ei ) 2 E χ v2iStatistical tablesThe following statistical tables are required for A Level Further Mathematics:Binomial Cumulative Distribution Function (see page 29)Poisson Cumulative Distribution Function (see page 35)Percentage Points of theχ2 Distribution (see page 36)Critical Values for Correlation Coefficients: Product Moment Coefficient and Spearman’s Coefficient(see page 37)Random Numbers (see page 38)Percentage Points of Student’sPercentage Points of the26t Distribution (see page 39)F Distribution (see page 40)Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further MathematicsMathematical Formulae and Statistical Tables – Issue 1 – July 2017 – Pearson Education Limited 2017

MPHMechanicsCentres of massFor uniform bodies:Triangular lamina:23along median from vertexCircular arc, radius r, angle at centre2α :Sector of circle, radius r, angle at centreSolid hemisphere, radiusfrom centreh: 14 h above the base on the line from centre of base to vertexh: 13 h above the base on the line from centre of base to vertexMotion in a circle.v rθTransverse acceleration:Radial acceleration:2r sin α3αr: 12 r from centre Solid cone or pyramid of heightTransverse velocity:2α :from centrer: 83 r from centreHemispherical shell, radius Conical shell of heightr sin αα.v rθ–rθ 2 –v2r4Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further MathematicsMathematical Formulae and Statistical Tables – Issue 1 – July 2017 – Pearson Education Limited 201727

MPH28Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further MathematicsMathematical Formulae and Statistical Tables – Issue 1 – July 2017 – Pearson Education Limited 2017

MPH5 Statistical TablesBinomial Cumulative Distribution FunctionThe tabulated value isp n 5, x 01234n 6, x 012345n 7, x 0123456n 8, x 01234567n 9, x 012345678n 10, x 0123456789P(X x), where X has a binomial distribution with index n and parameter n Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further MathematicsMathematical Formulae and Statistical Tables – Issue 1 – July 2017 – Pearson Education Limited 99905

MPHp n 12, x 01234567891011n 15, x 01234567891011121314n 20, x 99400.99

Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical Formulae and Statistical Tables Issue 1 uly 2017 Pearson Education Limited 2017. Standard deviation S. xx. n. or . x n x. 2. 2. . Statistical tables. The following statistical tables are required for A Level Mathematics: