Appendix B: Mathematical Formulae and Statistical Tables

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1 Aedi B: Mathematical Formulae ad Statistical Tables Pure Mathematics Mesuratio Surface area of shere = π r Area of curved surface of coe = π r slat height Trigoometry a = b + c bccosa Arithmetic Series u = a+ ( ) d S = ( a+ l) = { a+ ( ) d} Geometric Series u = ar S a( r ) = r a S = for r r < Summatios 6 r= r = ( + )(+ ) r = ( + ) r= OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 95 Oford, Cambridge ad RSA Eamiatios GCE Mathematics

2 Biomial Series + + = r r+ r+ r r ( a+ b) = a + a b+ a b + + a b + + b r ( ),! where = Cr = r r!( r)! ( ) ( ) ( r+ ) r ( + ) = r ( <, ) Logarithms ad Eoetials l e a = a Comle Numbers { r(cosθ + isi θ)} = r (cos θ + isi θ) iθ e = cosθ + isiθ The roots of z = are give by π k i z = e, for k = 0,,,, Maclauri s Series ( r) f( ) = f(0) + f (0) + f (0) + + f (0) +! r! r = = + + +! e e( )! r + + for all r r+ l( + ) = + + ( ) + ( < ) r r 5 r+ r si = + + ( ) + for all! 5! ( r + )! r r cos = + + ( ) + for all!! ( r )! 96 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios

3 ta 5 r+ r = + + ( ) + ( ) 5 r + 5 r+ sih = for all! 5! (r + )! r cosh = for all!! ( r)! tah 5 r+ = r + ( < < ) Hyerbolic Fuctios cosh sih = sih = sih cosh cosh = cosh + sih { } cosh = l + ( ) { } sih = l + + tah l + = ( ) < Coordiate Geometry The eredicular distace from ( h, k ) to a + by + c = 0 is ah + bk + c a + b The acute agle betwee lies with gradiets m ad m is ta m m + mm OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 97 Oford, Cambridge ad RSA Eamiatios GCE Mathematics

4 Trigoometric Idetities si( A± B) = si Acos B± cos Asi B cos( A± B) = cos Acos B si Asi B ta A± ta B ta( A± B) = ( A± B ( k + ) π ) taatab For t = ta A: t si A =, + t t cos A = + t A+ B A B si A+ si B= si cos A+ B A B si A si B= cos si A+ B A B cos A+ cos B= cos cos A+ B A B cos A cos B= si si Vectors The resolved art of a i the directio of b is a.b b The oit dividig AB i the ratio λ : µ is µ a+ λ b λ + µ Vector roduct: i a b ab ab ˆ a b= a b si θ = j a b = ab ab k a b ab ab If A is the oit with ositio vector a= ai+ aj+ ak ad the directio vector b is give by b= bi+ b j+ b k, the the straight lie through A with directio vector b has cartesia equatio a y a z a = = ( = λ) b b b 98 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios

5 The lae through A with ormal vector = i+ j+ k has cartesia equatio + y+ z + d= 0 where d= a. The lae through o-colliear oits A, B ad C has vector equatio r= a+ λ ( b a) + µ ( c a) = ( λ µ ) a+ λ b+ µ c The lae through the oit with ositio vector a ad arallel to b ad c has equatio r= a+ sb+ tc The eredicular distace of ( α, β, γ ) from + y+ z + d= 0 is α + β + γ + d + + Matri Trasformatios Aticlockwise rotatio through θ about O : cosθ siθ si θ cosθ Reflectio i the lie y = (ta θ ) : cos θ si θ si θ cosθ OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 99 Oford, Cambridge ad RSA Eamiatios GCE Mathematics

6 Differetiatio f( ) f( ) ta k si ksec k cos ta + sec sec ta cot cosec sih cosh tah sih cosh tah cosec cosec cot cosh sih sech + 00 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios

7 Itegratio (+ costat; a > 0 where relevat) f( ) f( )d sec k ta cot ta k k l sec l si cosec sec sih cosh tah l cosec + cot = l ta( ) l sec + ta = l ta( + π ) cosh sih l cosh a si ( < a) a a + a ta a a cosh or l { + a } ( > a) a a + a sih or l{ + + a } a a+ l = tah ( < a) a a a a a a l a + a dv du u d= uv v d d d OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 0 Oford, Cambridge ad RSA Eamiatios GCE Mathematics

8 Area of a Sector d A= r θ (olar coordiates) dy d A= y dt (arametric form) dt dt Numerical Mathematics Numerical Itegratio b The traezium rule: y d {( h y 0 + y ) + ( y a + y + + y ) }, where b a h = b Simso s rule: y d {( h y 0 + y ) + ( y a + y + + y ) + ( y + y + + y ) }, where b a h = ad is eve Numerical Solutio of Equatios f( ) The Newto-Rahso iteratio for solvig f( ) = 0 : + = f( ) Mechaics Motio i a Circle Trasverse velocity: v= rθ Trasverse acceleratio: v = rθ Radial acceleratio: v rθ = r 0 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios

9 Cetres of Mass For uiform bodies Triagular lamia: alog media from verte Solid hemishere, radius r: r from cetre Hemisherical shell, radius r: r from cetre 8 Circular arc, radius r, agle at cetre si α : r α from cetre α Sector of circle, radius r, agle at cetre r siα α : from cetre α Solid coe or yramid of height h: h above the base o the lie from cetre of base to verte Coical shell of height h: h above the base o the lie from cetre of base to verte Momets of Iertia For uiform bodies of mass m Thi rod, legth l, about eredicular ais through cetre: ml Rectagular lamia about ais i lae bisectig edges of legth l : ml Thi rod, legth l, about eredicular ais through ed: ml Rectagular lamia about edge eredicular to edges of legth l : ml Rectagular lamia, sides a ad b, about eredicular ais through cetre: ma ( + b ) Hoo or cylidrical shell of radius r about ais: mr Hoo of radius r about a diameter: mr Disc or solid cylider of radius r about ais: mr Disc of radius r about a diameter: mr OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 0 Oford, Cambridge ad RSA Eamiatios GCE Mathematics

10 Solid shere, radius r, about a diameter: 5 mr Sherical shell of radius r about a diameter: mr Parallel aes theorem: I = I + m( AG) A G Peredicular aes theorem: I z = I + Iy (for a lamia i the -y lae) Probability ad Statistics Probability P( A B) = P( A) + P( B) P( A B) P( A B) = P( A)P( B A) P( B A)P( A) P( A B) = P( B A)P( A) + P( B A )P( A ) Bayes Theorem: P( Aj)P( B Aj) P( Aj B) = Σ P( A)P( B A) i i Discrete Distributios For a discrete radom variable X takig values i with robabilities Eectatio (mea): E( X ) = µ =Σ i i i Variace: i i i i Var( X) = σ =Σ( µ ) =Σ µ For a fuctio g( X ) : E(g( X )) =Σg( ) The robability geeratig fuctio of X is G X ( t ) = E( t ), ad E( X ) = G (), X i i X Var( X ) = G () + G () {G ()} X X X For Z = X + Y, where X ad Y are ideedet: G ( t) = G ( t)g ( t) Z X Y 0 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios

11 Stadard Discrete Distributios: Distributio of X P(X = ) Mea Variace P.G.F. Biomial B(, ) ( ) ( ) ( + t) Poisso Po( λ) λ λ e λ λ ( ) e λ t! Geometric Geo () o,, ( ) t ( ) t Cotiuous Distributios For a cotiuous radom variable X havig robability desity fuctio f Eectatio (mea): E( X ) µ f( )d = = Variace: Var( X) = σ = ( µ ) f( )d= f( )d µ For a fuctio g( X ) : E(g( X )) = g( )f( )d Cumulative distributio fuctio: F( ) P( X ) f( t) dt = = tx The momet geeratig fuctio of X is M X ( t ) = E(e ) ad E( X ) = M (0), X ( ) X E( X ) = M (0), Var( X ) = M (0) {M (0)} X X For Z = X + Y, where X ad Y are ideedet: M ( t) = M ( t)m ( t) Z X Y OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 05 Oford, Cambridge ad RSA Eamiatios GCE Mathematics

12 Stadard Cotiuous Distributios: Distributio of X P.D.F. Mea Variace M.G.F. Uiform (Rectagular) o [ a, b] b a ( a b) ( b a) e e ( b a) t + bt at Eoetial λ λ e λ λ λ λ t Normal µ µ σ ( ) σ N(, ) e σ π µ σ e µ t+ σ t Eectatio Algebra Covariace: Cov( X, Y) = E(( X µ )( Y µ )) = E( XY) µ µ X Y X Y Var( ax ± by ) = a Var( X ) + b Var( Y ) ± abcov( X, Y ) Product momet correlatio coefficiet: Cov( X, Y ) ρ = σ σ X Y If X = ax + b ad Y = cy + d, the Cov( X, Y) = accov( X, Y ) For ideedet radom variables X ad Y E( XY) = E( X)E( Y) Var( ax ± by ) = a Var( X ) + b Var( Y ) Samlig Distributios For a radom samle X, X,, X of ideedet observatios from a distributio havig mea µ ad variace σ X is a ubiased estimator of µ, with Var( X ) σ = S is a ubiased estimator of σ, where Σ( X i X ) S = 06 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios

13 For a radom samle of observatios from X µ ~N(0, ) σ / N( µ, σ ) X µ ~ t S/ (also valid i matched-airs situatios) If X is the observed umber of successes i ideedet Beroulli trials i each of which the X robability of success is, ad Y =, the E( Y) = ad Var( Y ) = ( ) For a radom samle of observatios from y observatios from y N( µ, σ ) y N( µ, σ ) ad, ideedetly, a radom samle of ( X Y) ( µ µ y) ~N(0, ) σ σ y + y y If σ = σ = σ (ukow) the ( X Y) ( µ µ y) ~ t S + y + y, where + y y ( ) S ( ) S S = + y Correlatio ad Regressio For a set of airs of values (, y ) i i ( Σi ) S =Σ( i ) =Σi ( Σyi ) Syy =Σ( yi y) =Σyi ( Σi)( Σyi) Sy =Σ( i )( yi y) =Σiyi OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 07 Oford, Cambridge ad RSA Eamiatios GCE Mathematics

14 The roduct momet correlatio coefficiet is ( Σi)( Σyi) S Σy ( )( ) i i y Σ i yi y r = = = S Syy { Σ( i ) }{ Σ( yi y) } ( Σi) ( Σyi) Σi Σyi Searma s rak correlatio coefficiet is 6Σd rs = ( ) The regressio coefficiet of y o is S Σ( )( y y) b = = S y i i Σ( i ) Least squares regressio lie of y o is y = a+ b where a= y b Distributio-free (No-arametric) Tests Goodess-of-fit test ad cotigecy tables: ( O i E i ) E i ν ~ χ Aroimate distributios for large samles: Wilcoo Siged Rak test: T ~ N ( ( + ), ( + )(+ ) ) Wilcoo Rak Sum test (samles of sizes m ad, with m ): ( ) W ~ N m( m ), m( m ) 08 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios

15 OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 09 Oford, Cambridge ad RSA Eamiatios GCE Mathematics / / / / / / / / / / / / / / / / CUMULATIVE BINOMIAL PROBABILITIES = 5 = 0 5 = 6 = = 7 = = 8 =

16 0 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios / / / / / / / / / / / / CUMULATIVE BINOMIAL PROBABILITIES = 9 = = 0 = = =

17 OCR00 Aedi B: Mathematical Formulae ad Statistical Tables Oford, Cambridge ad RSA Eamiatios GCE Mathematics / / / / / / / / CUMULATIVE BINOMIAL PROBABILITIES = = = 6 =

18 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios / / / / / / / / CUMULATIVE BINOMIAL PROBABILITIES = 8 = = 0 =

19 OCR00 Aedi B: Mathematical Formulae ad Statistical Tables Oford, Cambridge ad RSA Eamiatios GCE Mathematics / / / / CUMULATIVE BINOMIAL PROBABILITIES = 5 =

20 Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios / / / / CUMULATIVE BINOMIAL PROBABILITIES = 0 =

21 CUMULATIVE POISSON PROBABILITIES λ = λ = λ = λ = OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 5 Oford, Cambridge ad RSA Eamiatios GCE Mathematics

22 CUMULATIVE POISSON PROBABILITIES λ = λ = Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios

23 CUMULATIVE POISSON PROBABILITIES λ = OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 7 Oford, Cambridge ad RSA Eamiatios GCE Mathematics

24 CUMULATIVE POISSON PROBABILITIES λ = Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios

25 THE NORMAL DISTRIBUTION FUNCTION If Z has a ormal distributio with mea 0 ad variace the, for each value of z, the table gives the value of Φ ( z), where: Φ ( z) = P( Z z) For egative values of z use Φ ( z) = Φ ( z). z ADD Critical values for the ormal distributio If Z has a ormal distributio with mea 0 ad variace the, for each value of, the table gives the value of z such that: P( Z z) = z OCR00 Aedi B: Mathematical Formulae ad Statistical Tables 9 Oford, Cambridge ad RSA Eamiatios GCE Mathematics

26 0//0 DRAFT MATHEMATICS DRAFT 0//0 CRITICAL VALUES FOR THE t-distribution If T has a t-distributio with ν degrees of freedom the, for each air of values of ad ν, the table gives the value of t such that: P( T t) = ν = Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios

27 CRITICAL VALUES FOR THE χ -DISTRIBUTION If X has a χ -distributio with ν degrees of freedom the, for each air of values of ad ν, the table gives the value of such that: P( X ) = ν = OCR00 Aedi B: Mathematical Formulae ad Statistical Tables Oford, Cambridge ad RSA Eamiatios GCE Mathematics

28 WILCOXON SIGNED RANK TEST P is the sum of the raks corresodig to the ositive differeces, Q is the sum of the raks corresodig to the egative differeces, T is the smaller of P ad Q. For each value of the table gives the largest value of T which will lead to rejectio of the ull hyothesis at the level of sigificace idicated. Critical values of T Level of sigificace Oe Tail Two Tail = For larger values of, each of P ad Q ca be aroimated by the ormal distributio with mea + ( ) ad variace ( + )(+ ). Aedi B: Mathematical Formulae ad Statistical Tables OCR 00 GCE Mathematics Oford, Cambridge ad RSA Eamiatios

29 WILCOXON RANK SUM TEST The two samles have sizes m ad, where m. R m is the sum of the raks of the items i the samle of size m. W is the smaller of R m ad m ( + m+ ) Rm. For each air of values of m ad, the table gives the largest value of W which will lead to rejectio of the ull hyothesis at the level of sigificace idicated. Critical values of W Level of sigificace Oe Tail Two Tail m = m = m = 5 m = Level of sigificace Oe Tail Two Tail m = 7 m = 8 m = 9 m = For larger values of m ad, the ormal distributio with mea mm ( + + ) ad variace m( m + + ) should be used as a aroimatio to the distributio of R m. OCR00 Aedi B: Mathematical Formulae ad Statistical Tables Oford, Cambridge ad RSA Eamiatios GCE Mathematics