Appendix A: Mathematical Formulae and Statistical Tables

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1 Aedi A: Mathematical Formulae ad Statistical Tables Pure Mathematics Mesuratio Surface area of shere = r Area of curved surface of coe = r J slat height Trigoometry a * b & c ' bc cos A Arithmetic Series u * a & ( ' ) d S * ( a & l) * { a & ( ' ) d} Geometric Series u * ar ' S a( ' r ) * ' r a SE * for r ' r ) Summatios _ 6 r* r * ( & )( & ) r * ( & ) r* _ 8 Aedi A: Mathematical Formulae ad Statistical Tables OCR 0

2 Biomial Series ` j ` j ` & j a k & a k * a k b r l b r & l b r & l ` j ' ` j ' ` j ' r r ( a & b) * a & a k a b & a k a b & & a b & & b a k r b l b l b l ( W ) ` j! where a k * Cr * b r l r!( ' r)! r ( ' ) ( ' ) "( ' r & ) ( & ) * & & & " &..." r & " ( ), W' ) Logarithms ad Eoetials l e a * a Comle Numbers { r(cos & isi )} * r (cos & isi ) i e * cos & isi The roots of z * are give by k i z * e, for k * 0,,,, ' Maclauri s Series ( r) f( ) * f(0) & f C(0) & f CC(0) & & f (0) &! r! r r * * & & & &! e e( )! r & for all r& l( & ) * ' & ' &(' ) & (' ) D ) r r 5 r& r si * ' & ' & (' ) & for all! 5! ( r & )! r cos * ' & ' & (' ) & for all!! ( r )! r OCR 0 Aedi A: Mathematical Formulae ad Statistical Tables 85

3 ta ' 5 * ' & ' &(' ) 5 r& r & (' D D) r & 5 r& sih * & & & & & for all! 5! (r & )! r cosh * & & & & & for all!! ( r)! tah ' 5 r& * & & & & & 5 r & (' ) ) ) Hyerbolic Fuctios cosh ' sih * sih * sih cosh cosh * cosh & sih { } ' cosh * l & ' ( I) { } ' sih * l & & ' `& j tah * l ( ) a k ) b' l Coordiate Geometry The eredicular distace from ( h, k) to a & by & c * 0 is The acute agle betwee lies with gradiets m ad m is ta ah & bk & c a & b m ' m & m m ' 86 Aedi A: Mathematical Formulae ad Statistical Tables OCR 0

4 Trigoometric Idetities si( A H B) * si Acos B H cos Asi B cos( A H B) * cos Acos B $ si Asi B ta A H ta B ta( A H B) * ( A H B O ( k & ) ) $ ta Ata B 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 : is a & & b Vector roduct: i a b ` ab ' ab j ˆ a k aj b * a b si * j a b * ab ' ab k a b a ab ' ab k b l If A is the oit with ositio vector a * ai & aj & ak ad the directio vector b is give by b * b i & b j & b k, the the straight lie through A with directio vector b has cartesia equatio ' a y ' a z ' a * * (* ) b b b OCR 0 Aedi A: Mathematical Formulae ad Statistical Tables 87

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 & ( b ' a) & ( c ' a) * ( ' ' ) a & b & 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 (,, ) from & y & z & d * 0 is & & & d & & Matri Trasformatios Aticlockwise rotatio through about O : ` cos a b si ' si cos j k l Reflectio i the lie y * (ta ) : `cos si a b si ' cos j k l 88 Aedi A: Mathematical Formulae ad Statistical Tables OCR 0

6 Differetiatio f( ) f C( ) ta k si ' k sec k ' cos ' ' ' ta ' & sec sec ta cot cosec sih cosh tah sih cosh tah ' ' ' 'cosec ' cosec cot cosh sih sech & ' ' If f( ) y * the g( ) dy f C( )g( ) ' f( )g C( ) * d {g( )} OCR 0 Aedi A: Mathematical Formulae ad Statistical Tables 89

7 Itegratio (+ costat; a + 0 where relevat) f( ) f( )d f sec k ta cot ta k k l sec l si cosec ' l cosec & cot * l ta( ) sec l sec & ta * l ta( & ) sih cosh tah cosh sih l cosh a ' ' ` j si a k ( ) a) b a l a & ' a ta ' ` j a k a b a l cosh ' ` j a k or l { & ' a } ( + a) b a l a & a k b a l sih ' ` j or l{ & & a } a ' a & ' ` j l * tah a k ( ) a) a a ' a b a l ' a ' a l a & a f f dv du u d * uv ' v d d d 90 Aedi A: Mathematical Formulae ad Statistical Tables OCR 0

8 Area of a Sector r d A * f (olar coordiates) f ` dy d j A * y dt a ' k b dt dt l (arametric form) Numerical Mathematics Numerical Itegratio b The traezium rule: f y d Q h {( y 0 & y ) & ( y a & y & & y ) ' }, where b ' a h * b Simso s rule: f y d Q 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 C( ) Mechaics Motio i a Circle Trasverse velocity: v * r! Trasverse acceleratio: v! * r!! Radial acceleratio: v ' r! * ' r OCR 0 Aedi A: Mathematical Formulae ad Statistical Tables 9

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 Circular arc, radius r, agle at cetre 8 si : r from cetre Sector of circle, radius r, agle at cetre r si : from cetre 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: m( a & 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 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 & I y (for a lamia i the -y lae) 9 Aedi A: Mathematical Formulae ad Statistical Tables OCR 0

10 Probability ad Statistics Probability P( A T B) * P( A) & P( B) ' P( A S B) P( A S B) * P( A)P( B A) P( B A)P( A) P( A B) * P( B A)P( A) & P( B AC)P( AC) Bayes Theorem: P( Aj ) P( B Aj ) P( Aj B) * / P( A ) P( B A ) i i Discrete Distributios For a discrete radom variable X takig values Eectatio (mea): E( X ) * * / i i i with robabilities i Variace: i i i i Var( X ) * * /( ' ) * / ' For a fuctio g( X ) : E(g( X )) * / g( ) The robability geeratig fuctio of X is G X ( t ) * E( t ), ad E( X ) * G C (), X i i X Var( X ) * G CC () & G C () '{G C ()} X X X For Z * X & Y, where X ad Y are ideedet: G ( t) * G ( t)g ( t) Z X Y OCR 0 Aedi A: Mathematical Formulae ad Statistical Tables 9

11 Stadard Discrete Distributios: Distributio of X P(X = ) Mea Variace P.G.F. ` j Biomial B(, ) a k ( ' ) b l ' ( ' ) ( ' & t) Poisso Po( ) e ' ( ) e t'! Geometric Geo () o,, ( ' ) ' ' t ' ( ' ) t Cotiuous Distributios For a cotiuous radom variable X havig robability desity fuctio f Eectatio (mea): E( X ) * * f f( )d Variace: f Var( X ) * * ( ' ) f( )d * f( )d ' f For a fuctio g( X ) : E(g( X )) * f g( )f( )d Cumulative distributio fuctio: F( ) * P( X D ) * f f( t) dt 'E tx The momet geeratig fuctio of X is M X ( t ) * E(e ) ad E( X ) * M C (0), X ( ) X E( X ) * M (0), Var( X ) * M CC (0) '{M C (0)} X X For Z * X & Y, where X ad Y are ideedet: M ( t) * M ( t)m ( t) Z X Y 9 Aedi A: Mathematical Formulae ad Statistical Tables OCR 0

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 e ' ' t Normal N(, ) e ' ' e t& t Eectatio Algebra Covariace: Cov( X, Y ) * E(( X ' )( Y ' )) * E( XY ) ' X Y X Y Var( ax H by ) * a Var( X ) & b Var( Y ) H abcov( X, Y ) Product momet correlatio coefficiet: Cov( X, Y ) * X Y If X * ax C & b ad Y * cyc & d, the Cov( X, Y ) * ac Cov( X C, YC) For ideedet radom variables X ad Y E( XY) * E( X )E( Y ) Var( ax H by ) * a Var( X ) & b Var( Y ) Samlig Distributios For a radom samle X, X,, X of ideedet observatios from a distributio havig mea ad variace X is a ubiased estimator of, with Var( X ) * S is a ubiased estimator of, where /( Xi ' X ) S * ' OCR 0 Aedi A: Mathematical Formulae ad Statistical Tables 95

13 For a radom samle of observatios from N(, ) X ' / ~ N(0, ) X ' S / ~ t ' (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 y observatios from observatios from y N(, ) y N(, ) ad, ideedetly, a radom samle of ( X ' Y ) ' ( ' y ) ~ N(0, ) & y y If y * * (ukow) the ( X ' Y ) ' ( ' y ) ~ ` j S & a k b y l t & y ', where ' & y ' y ( ) S ( ) S S * & ' y Correlatio ad Regressio For a set of airs of values (, y ) i i (/ i ) S * /( i ' ) * / i ' (/ yi ) S yy * /( yi ' y) * / yi ' (/ i )(/ yi ) Sy * /( i ' )( yi ' y) * / i yi ' 96 Aedi A: Mathematical Formulae ad Statistical Tables OCR 0

14 The roduct momet correlatio coefficiet is (/ i )(/ yi ) S / ( )( ) i yi ' y / i ' yi ' y r * * * S S yy {/( i ' ) }{/( yi ' y) } ` (/ i ) j ` (/ yi ) j a / i ' / yi ' k a k b l b l Searma s rak correlatio coefficiet is 6/ d rs * ' ( ' ) The regressio coefficiet of y o is S /( ' )( y ' y) b * * S y i i /( 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 ~ Aroimate distributios for large samles: Wilcoo Siged Rak test: T ~ N ( & ), ( & )( & ) Wilcoo Rak Sum test (samles of sizes m ad, with m D ): W ~ N m( m & & ), m( m & & ) OCR 0 Aedi A: Mathematical Formulae ad Statistical Tables 97

15 98 Aedi A: Mathematical Formulae ad Statistical Tables OCR 0 CUMULATIVE BINOMIAL PROBABILITIES / / / / / / / / / / / / / / / / = 5 = 0 5 = 6 = = 7 = = 8 =

16 OCR 0 Aedi A: Mathematical Formulae ad Statistical Tables 99 CUMULATIVE BINOMIAL PROBABILITIES / / / / / / / / / / / / = 9 = = 0 = = =

17 00 Aedi A: Mathematical Formulae ad Statistical Tables OCR 0 CUMULATIVE BINOMIAL PROBABILITIES / / / / / / / / = = = 6 =

18 OCR 0 Aedi A: Mathematical Formulae ad Statistical Tables 0 CUMULATIVE BINOMIAL PROBABILITIES / / / / / / / / = 8 = = 0 =

19 0 Aedi A: Mathematical Formulae ad Statistical Tables OCR 0 CUMULATIVE BINOMIAL PROBABILITIES / / / / = 5 =

20 OCR 0 Aedi A: Mathematical Formulae ad Statistical Tables 0 CUMULATIVE BINOMIAL PROBABILITIES / / / / = 0 =

21 CUMULATIVE POISSON PROBABILITIES = = = = Aedi A: Mathematical Formulae ad Statistical Tables OCR 0

22 CUMULATIVE POISSON PROBABILITIES = = OCR 0 Aedi A: Mathematical Formulae ad Statistical Tables 05

23 CUMULATIVE POISSON PROBABILITIES = Aedi A: Mathematical Formulae ad Statistical Tables OCR 0

24 CUMULATIVE POISSON PROBABILITIES = OCR 0 Aedi A: Mathematical Formulae ad Statistical Tables 07

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.( z), where:.( z) * P( Z D z) For egative values of z use.(' z) * '.( 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 D z) * z Aedi A: Mathematical Formulae ad Statistical Tables OCR 0

CRITICAL VALUES FOR THE t-distribution If T has a t-distributio with degrees of freedom the, for each air of values of ad, the table gives the value of t such that: P( T D t) *. 0.75 0.90 0.95 0.

26 CRITICAL VALUES FOR THE t-distribution If T has a t-distributio with degrees of freedom the, for each air of values of ad, the table gives the value of t such that: P( T D t) * = E OCR 0 Aedi A: Mathematical Formulae ad Statistical Tables 09

CRITICAL VALUES FOR THE -DISTRIBUTION If X has a -distributio with degrees of freedom the, for each air of values of ad of such that: P( X D ) *, the table gives the value 0.0 0.05 0.05 0.9 0.95 0.

27 CRITICAL VALUES FOR THE -DISTRIBUTION If X has a -distributio with degrees of freedom the, for each air of values of ad of such that: P( X D ) *, the table gives the value = Aedi A: Mathematical Formulae ad Statistical Tables OCR 0

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 ( & )( & ). OCR 0 Aedi A: Mathematical Formulae ad Statistical Tables

29 WILCOXON RANK SUM TEST The two samles have sizes m ad, where m D. Rm is the sum of the raks of the items i the samle of size m. W is the smaller of Rm 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 m( m & & ) ad variace m( m & & ) should be used as a aroimatio to the distributio of R m. Aedi A: Mathematical Formulae ad Statistical Tables OCR 0

Notation List. For Cambridge International Mathematics Qualifications. For use from 2020

Notation List. For Cambridge International Mathematics Qualifications. For use from 2020 Notatio List For Cambridge Iteratioal Mathematics Qualificatios For use from 2020 Notatio List for Cambridge Iteratioal Mathematics Qualificatios (For use from 2020) Mathematical otatio Eamiatios for CIE