mathematics_book_for_isi_exam.pdf - MATHEMATICS BOOK FOR ISI ENTRANCES Ctanujit Classes The following Syllabi are covered here ISI MSTAT PSA(OBJECTIVE

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Unformatted text preview: MATHEMATICS BOOK FOR ISI ENTRANCES Ctanujit Classes The following Syllabi are covered here: ISI MSTAT PSA (OBJECTIVE TYPE) SYLLABUS 2017 ISI MSQMS QMA & QMS (OBJECTIVE & SUBJECTIVE TYPE) SYLLABUS 2017 ISI MMATH, MTECH QROR & MTECH CS MMA (OBJECTIVE TYPE) 2017 SYLLABUS TABLE OF CONTENTS: Formulae at a Glance (Page No. 5 – 20) Permutation & Combination (Page No. 21 – 32) Inequality & Number Theory (Page No. 33 – 60) Polynomials (Page No. 61 – 77) Combinatorial Probability (Page No. 78 – 92) Complex Number & Theory of Equations (Page No. 93 – 104) Set, Functions & Matrices (Page No. 105 – 128) Coordinate Geometry (Page No. 129 – 168) Functions & Graphs (Page No. 169 – 242) Differential Calculus (Page No. 243 – 286) Integral Calculus (Page No. 287 – 300) Ctanujit Classes Page No. 3 Arithmetic series General (kth) term, last (nth) term, l = Sum to n terms, Geometric series General (kth) term, Sum to n terms, Sum to infinity Infinite series f(x) uk = a + (k – 1)d un = a + (n – l)d Sn = 1–2 n(a + l) = 1–2 n[2a + (n – 1)d] uk = a r k–1 a(1 – r n) a(r n – 1) Sn = –––––––– = –––––––– 1–r r–1 x2 xr = f(0) + xf'(0) + –– f"(0) + ... + –– f (r)(0) + ... 2! r! f(x) (x – a)2 (x – a)rf(r)(a) = f(a) + (x – a)f'(a) + –––––– f"(a) + ... + –––––––––– + ... r! 2! f(a + x) x2 xr = f(a) + xf'(a) + –– f"(a) + ... + –– f(r)(a) + ... 2! r! x2 xr ex = exp(x) = 1 + x + –– + ... + –– + ... , all x 2! r! a ,–1<r<1 S∞ = ––––– 1–r x2 x3 xr = x – –– + –– – ... + (–1)r+1 –– + ... , – 1 < x < 1 2 3 r sin x x3 x5 x 2r+1 = x – –– + –– – ... + (–1)r –––––––– + ... , all x 3! 5! (2r + 1)! cos x x2 x4 x 2r = 1 – –– + –– – ... + (–1)r –––– + ... , all x 2! 4! (2r)! arctan x x3 x5 x 2r+1 = x – –– + –– – ... + (–1)r –––––– + ... , – 1 < x < 1 3 5 2r + 1 General case sinh x n(n – 1) 2 n(n – 1) ... (n – r + 1) ––––––––––––––––– xr + ... , |x| < 1, (1 + x)n = 1 + nx + ––––––– x + ... + 2! 1.2 ... r n∈R x3 x5 x 2r+1 = x + –– + –– + ... + –––––––– + ... , all x 3! 5! (2r + 1)! cosh x x2 x4 x 2r = 1 + –– + –– + ... + –––– + ... , all x 2! 4! (2r)! artanh x x3 x5 x 2r+1 = x + –– + –– + ... + –––––––– + ... , – 1 < x < 1 3 5 (2r + 1) Binomial expansions When n is a positive integer () () () n n n (a + b)n = an + 1 an – 1 b + 2 an –2 b2 + ... + r an–r br + ... bn , n ∈ N where n n n n+1 n! n + r+1 = r+1 r = Cr = –––––––– r r!(n – r)! () () ( ) ( ) 2 Logarithms and exponentials exln a = ax logbx loga x = ––––– logba Numerical solution of equations f(xn) Newton-Raphson iterative formula for solving f(x) = 0, xn+1 = xn – –––– f'(xn) Complex Numbers {r(cos θ + j sin θ)}n = r n(cos nθ + j sin nθ) ejθ = cos θ + j sin θ 2πk j) for k = 0, 1, 2, ..., n – 1 The roots of zn = 1 are given by z = exp( –––– n Finite series n n 1 1 ∑ r2 = – n(n + 1)(2n + 1) ∑ r3 = – n2(n + 1)2 4 6 r=1 r=1 ALGEBRA ln(1 + x) Hyperbolic functions cosh2x – sinh2x = 1, sinh2x = 2sinhx coshx, cosh2x = cosh2x + sinh2x arcosh x = ln(x + x 2 + 1 ), 1 + x 1 artanh x = –2 ln ––––– 1 – x , |x| < 1 arsinh x = ln(x + ( x 2 – 1 ), x > 1 ) Matrices Anticlockwise rotation through angle θ, centre O: Reflection in the line y = x tan θ : Ctanujit Classes θ ( cos sin θ 2θ ( cos sin 2θ –sin θ cos θ ) sin 2θ –cos 2θ ) Page No. 5 Cosine rule A b2 + c2 – a2 (etc.) cos A = –––––––––– 2bc c a2 = b2 + c2 –2bc cos A (etc.) B Trigonometry Perpendicular distance of a point from a line and a plane b a Line: (x1,y1) from ax + by + c = 0 : a2 + b2 n1α + n2β + n3γ + d Plane: (α,β,γ) from n1x + n2y + n3z + d = 0 : –––––––––––––––––– √(n12 + n22 + n32) C sin (θ ± φ) = sin θ cos φ ± cos θ sin φ cos (θ ± φ) = cos θ cos φ 7 sin θ sin φ Vector product i a1 b1 a2b3 – a3b2 a × b = |a| |b| sinθ ^n = j a2 b2 = a3b1 – a1b3 k a3 b3 a1b2 – a2b1 tan θ ± tan φ tan (θ ± φ) = –––––––––––– , [(θ ± φ) ≠ (k + W)π] 1 7 tan θ tan φ | a × (b × c) = (c . a) b – (a . b) c sin θ – sin φ = 2 cos 1–2 (θ + φ) sin 1–2 (θ – φ) Conics 1– 2 (θ – φ) 1– 2 (θ – φ) Ellipse Parabola Hyperbola Rectangular hyperbola Standard form y2 x2 + –– –– =1 b2 a2 y2 = 4ax y2 x2 – –– –– =1 b2 a2 x y = c2 Parametric form (acosθ, bsinθ) (at2, 2at) (asecθ, btanθ) (ct, –c–) t e<1 b2 = a2 (1 – e2) e=1 e>1 b2 = a2 (e2 – 1) e = √2 Foci (± ae, 0) (a, 0) (± ae, 0) (±c√2, ±c√2) Directrices x = ± –a e x = –a x = ± –ae x + y = ±c√2 Asymptotes none none y x a– = ± –b x = 0, y = 0 3 Vectors and 3-D coordinate geometry (The position vectors of points A, B, C are a, b, c.) The position vector of the point dividing AB in the ratio λ:µ µa + λb is ––––––– (λ + µ) Line: ) a1 b1 c1 a. (b × c) = a2 b2 c2 = b. (c × a) = c. (a × b) a3 b3 c3 (1 + t ) sin θ + sin φ = 2 sin 1–2 (θ + φ) cos 1–2 (θ – φ) cos θ + cos φ = 2 cos 1–2 (θ + φ) cos cos θ – cos φ = –2 sin 1–2 (θ + φ) sin | |( | Cartesian equation of line through A in direction u is x – a1 y – a2 z – a3 =t = –––––– = –––––– –––––– u1 u2 u3 ( ) Eccentricity a.u The resolved part of a in the direction u is ––––– |u| Any of these conics can be expressed in polar coordinates (with the focus as the origin) as: where l is the length of the semi-latus rectum. Plane: Cartesian equation of plane through A with normal n is n1 x + n2y + n3z + d = 0 where d = –a . n Mensuration The plane through non-collinear points A, B and C has vector equation r = a + s(b – a) + t(c – a) = (1 – s – t) a + sb + tc The plane through A parallel to u and v has equation r = a + su + tv Cone : TRIGONOMETRY, VECTORS AND GEOMETRY (1 – t2) 2t , cos θ = –––––– For t = tan 1–2 θ : sin θ = –––––– 2 2 (1 + t ) ax1 + by1 + c l – = 1 + e cos θ r Sphere : Surface area = 4π r2 Curved surface area = π r × slant height Ctanujit Classes Page No. 6 Differentiation f(x) tan kx sec x cot x cosec x arcsin x f'(x) ksec2 kx sec x tan x –cosec2 x –cosec x cot x 1 ––––––– √(1 – x2) –1 ––––––– √(1 – x2) 1 ––––– 1 + x2 cosh x sinh x sech2 x 1 ––––––– √(1 + x2) arccos x arctan x sinh x cosh x tanh x arsinh x artanh x 4 du dv v ––– – u ––– dx dx u dy Quotient rule y = – , ––– = 2 v v dx Trapezium rule b ––– a ∫a ydx ≈ 1–2 h{(y0 + yn) + 2(y1 + y2 + ... + yn –1)}, where h = ––– n b Integration by parts Area of a sector Arc length dv du dx = uv – ∫ v ––– dx ∫ u ––– dx dx A = 1–2 ∫ r dθ (polar coordinates) A = 1–2 ∫ (x˙y – yx˙) dt (parametric form) s = ∫ √ (x˙ + y˙ ) dt (parametric form) dy s = ∫ √ (1 + [ –––] ) dx (cartesian coordinates) dx dr s = ∫ √ (r + [ –––] ) dθ (polar coordinates) dθ 2 2 2 2 2 2 sec x 1 –––––– x2 – a2 1 ––––––– √(a2 – x2) 1 –––––– a2 + x2 1 –––––– a2 – x2 sinh x cosh x tanh x 1 ––––––– √(a2 + x2) 1 ––––––– √(x2 – a2) Surface area of revolution ∫f(x) dx (+ a constant) (l/k) tan kx ln |sec x| ln |sin x| x –ln |cosec x + cot x| = ln |tan – 2| x π ln |sec x + tan x| = ln tan – + – 2 4 1 x–a –– ln ––– x +–– a 2a | ( | | x arcsin ( – a ) , |x| < a 1 x – a arctan( – a) 1 1 x a+x –– ln ––––– = – artanh ( – a ) , |x| < a 2a | a – x | a cosh x sinh x ln cosh x x 2 2 arsinh – a or ln (x + x + a ), () x arcosh ( – a ) or ln (x + x S = 2π∫y ds = 2π∫y√(x˙ S = 2π∫x ds = 2π∫x√(x˙ 2 – a 2 ), x > a , a > 0 2 + y˙ 2) dt 2 + y˙ 2) dt x y Curvature )| CALCULUS 1 ––––––– √(x2 – 1) 1 ––––––– (1 – x2) arcosh x Integration f(x) sec2 kx tan x cot x cosec x d2y ––– dψ x˙ ÿ – x¨ y˙ dx2 κ = ––– = –––––––– 3/2 = ––––––––––––– 2 2 ds dy 2 3/2 (˙x + y˙ ) 1 + –– dx ( [ ]) 1 Radius of curvature ρ = –– κ, Centre of curvature c = r + ρ n^ L'Hôpital’s rule If f(a) = g(a) = 0 and g'(a) ≠ 0 then f(x) f'(a) Lim –––– = –––– x ➝a g(x) g'(a) Multi-variable calculus ∂g/∂x ∂w ∂w ∂w grad g = ∂g/∂y For w = g(x, y, z), δw = ––– δx + ––– δy + ––– δz ∂x ∂y ∂z ∂g/∂z ( ) Ctanujit Classes Page No. 7 Centre of mass (uniform bodies) Triangular lamina: Solid hemisphere of radius r: Hemispherical shell of radius r: Solid cone or pyramid of height h: Moments of inertia (uniform bodies, mass M) 2 – along median from vertex 3 3 – r from centre 8 1 – r from centre 2 1 – h above the base on the 4 line from centre of base to vertex Sector of circle, radius r, angle 2θ: 2r sin θ ––––––– from centre 3θ r sin θ from centre Arc of circle, radius r, angle 2θ at centre: ––––––– θ 1 – h above the base on the 3 line from the centre of Conical shell, height h: Thin rod, length 2l, about perpendicular axis through centre: Rectangular lamina about axis in plane bisecting edges of length 2l: Thin rod, length 2l, about perpendicular axis through end: Rectangular lamina about edge perpendicular to edges of length 2l: Rectangular lamina, sides 2a and 2b, about perpendicular 1 – M(a2 + b2) axis through centre: 3 Hoop or cylindrical shell of radius r about perpendicular axis through centre: Hoop of radius r about a diameter: Disc or solid cylinder of radius r about axis: base to the vertex Solid sphere of radius r about a diameter: 5 Motion in polar coordinates Spherical shell of radius r about a diameter: Transverse velocity: v = rθ˙ v2 Radial acceleration: –rθ˙ 2 = – –– r Transverse acceleration: v˙ = rθ¨ Mr2 1 – Mr2 2 1 – Mr2 2 1 – Mr2 4 2 – Mr2 5 2 – Mr2 3 Parallel axes theorem: IA = IG + M(AG)2 Perpendicular axes theorem: Iz = Ix + Iy (for a lamina in the (x, y) plane) MECHANICS Disc of radius r about a diameter: Motion in a circle 1 – Ml2 3 1 – Ml2 3 4 – Ml2 3 4 – Ml2 3 General motion Radial velocity: Transverse velocity: Radial acceleration: Transverse acceleration: r˙ rθ˙ r¨ – rθ˙ 2 1 d rθ¨ + 2r˙θ˙ = – –– (r2θ˙) r dt Moments as vectors The moment about O of F acting at r is r × F Ctanujit Classes Page No. 8 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') P(Aj)P(B|Aj) Bayes’ Theorem: P(A j |B) = –––––––––––– ∑P(Ai)P(B|Ai) Discrete distributions X is a random variable taking values xi in a discrete distribution with P(X = xi) = pi Expectation: µ = E(X) = ∑xi pi σ2 = Var(X) = ∑(xi – µ)2 pi = ∑xi2pi – µ2 Variance: For a function g(X): E[g(X)] = ∑g(xi)pi Rank correlation: Spearman’s coefficient 6 X is a continuous variable with probability density function (p.d.f.) f(x) Expectation: µ = E(X) = ∫ x f(x)dx Variance: σ2 = Var (X) = ∫(x – µ)2 f(x)dx = ∫x2 f(x)dx – µ2 For a function g(X): E[g(X)] = ∫g(x)f(x)dx Cumulative x distribution function F(x) = P(X < x) = ∫–∞f(t)dt For a sample of n pairs of observations (xi, yi) (∑xi)2 (∑yi)2 , , Syy = ∑(yi – y )2 = ∑yi2 – ––––– Sxx = ∑(xi – x )2 = ∑xi2 – ––––– n n (∑xi)(∑yi) Sxy = ∑(xi – x )(yi – y ) = ∑xi yi – ––––––––– n Sxy = ∑( xi – x )( yi – y ) = ∑ xi yi – x y –––– n n n Regression Least squares regression line of y on x: y – y = b(x – x ) ∑ xi yi –xy Sxy ∑(xi – x) (yi – y ) n = ––––––––––––––– = b = ––– Sxx ∑ xi 2 ∑(xi – x )2 – x2 n Estimates Unbiased estimates from a single sample σ2 X for population mean µ; Var X = –– n 1 ∑(x – x )2f S2 for population variance σ 2 where S2 = –––– i i n–1 STATISTICS Continuous distributions Covariance [ 6∑di2 rs = 1 – –––––––– n(n2 – 1) Populations Correlation and regression Product-moment correlation: Pearson’s coefficient ∑ xi yi –xy Sxy Σ( xi – x )( yi – y ) n r= = = 2 2 Sxx Syy ∑ xi 2 ∑ yi 2 Σ( xi – x ) Σ( yi – y ) − x2 − y2 n n Probability generating functions For a discrete distribution G(t) = E(tX) E(X) = G'(1); Var(X) = G"(1) + µ – µ2 GX + Y (t) = GX (t) GY (t) for independent X, Y Moment generating functions: MX(θ) = E(eθX) E(X) = M'(0) = µ; E(Xn) = M(n)(0) Var(X) = M"(0) – {M'(0)}2 MX + Y (θ) = MX (θ) MY (θ) for independent X, Y Ctanujit Classes Page No. 9 Markov Chains pn + 1 = pnP Long run proportion p = pP Bivariate distributions Covariance Cov(X, Y) = E[(X – µX)(Y – µY)] = E(XY) – µXµY Cov(X, Y) ρ = –––––––– Product-moment correlation coefficient σX σY Sum and difference Var(aX ± bY) = a2Var(X) + b2Var(Y) ± 2ab Cov (X,Y) If X, Y are independent: Var(aX ± bY) = a2Var(X) + b2Var(Y) E(XY) = E(X) E(Y) Coding X = aX' + b ⇒ Cov(X, Y) = ac Cov(X', Y') Y = cY ' + d } One-factor model: xij = µ + αi + εij, where εij ~ N(0,σ2) Ti2 T 2 SSB = ∑ni ( x i – x )2 = ∑ ––– ni – –– n i i SST = ∑ ∑ (xij – i j x )2 = ∑ ∑ xij i j 2 2 –– – T n Yi α + βxi + εi RSS ∑(yi – a – bxi)2 α + βf(xi) + εi α + βxi + γzi + εi ∑(yi – a – εi ~ N(0, σ2) No. of parameters, p 2 bf(xi))2 ∑(yi – a – bxi – 2 czi)2 3 a, b, c are estimates for α, β, γ. For the model Yi = α + βxi + εi, Sxy σ2 , b = ––– , b ~ N β, ––– Sxx Sxx ( b−β ~ tn – 2 σˆ 2 / Sxx ) σ 2∑xi2 a = y – b x , a ~ N α, ––––––– nS ( xx RSS σ^2 = n–––– –p ) (x0 – x )2 a + bx0 ~ N(α + βx0, σ2 1 + ––––––– – n Sxx 2 (Sxy) RSS = Syy – ––––– = Syy (1 – r2) Sxx { } Randomised response technique –y – (1 – θ) n E(p^) = –––––––––– (2θ – 1) [(2θ – 1) p + (1 – θ)][θ – (2θ – 1)p] STATISTICS 7 Analysis of variance Regression Var(p^) = –––––––––––––––––––––––––––––– 2 n(2θ – 1) Factorial design Interaction between 1st and 2nd of 3 treatments (–) { (Abc – abc) + (AbC – abC) (ABc – aBc) + (ABC – aBC) ––––––––––––––––––––– – –––––––––––––––––––––– 2 2 } Exponential smoothing y^n+1 = α yn + α(1 – α)yn–1 + α(1 – α)2 yn–2 + ... + α(1 – α)n–1 y1 + (1 – α)ny0 ^y = y^ + α(y – y^ ) n n n+1 n y^n+1 = α yn + (1 – α) y^n Ctanujit Classes Page No. 10 Description Pearson’s product moment correlation test r= Distribution ∑ xi yi –xy n ∑ xi 2 ∑ yi 2 – x2 – y2 n n t-test for the difference in the means of 2 samples 6∑di2 rs = 1 – ––––––– n(n2 – 1) 8 Normal test for a mean x–µ σ/ n N(0, 1) t-test for a mean x–µ s/ n tn – 1 χ2 test t-test for paired sample Normal test for the difference in the means of 2 samples with different variances Description ∑ (f o – fe ) ( x1 – x2 ) – µ s/ n N(0, 1) 1 + n2 – 2 (n1 – 1)s12 + (n2 – 1)s22 where s2 = ––––––––––––––––––––––– n1 + n2 – 2 See tables Wilcoxon Rank-sum (or Mann-Whitney) 2-Sample test Samples size m, n: m < n Wilcoxon W = sum of ranks of sample size m Mann-Whitney T = W 1–2 m(m + 1) See tables p –θ Normal test on binomial proportion θ (1 – θ ) n χ2 test for variance (n – 1)s 2 σ2 ( x – y ) – ( µ1 – µ2 ) σ 12 σ 2 2 + n1 n2 tn A statistic T is calculated from the ranked data. χ 2v t with (n – 1) degrees of freedom ( x – y ) – ( µ1 – µ2 ) 1 1 + s n1 n2 Distribution Wilcoxon single sample test 2 fe Test statistic F-test on ratio of two variances s12 /σ12 ––––––– s22 /σ22 Ctanujit Classes , s12 > s22 N(0, 1) STATISTICS: HYPOTHESIS TESTS Spearman rank correlation test Test statistic χ2n – 1 Fn 1 –1, n2 –1 Page No. 11 Function Name Mean Variance p.g.f. G(t) (discrete) m.g.f. M(θ) (continuous) P(X = r) = nCr qn–rpr , for r = 0, 1, ... ,n , 0 < p < 1, q = 1 – p np npq G(t) = (q + pt)n Poisson (λ) Discrete λr P(X = r) = e–λ ––– r! , for r = 0, 1, ... , λ > 0 λ λ G(t) = eλ(t – 1) µ σ2 M(θ) = exp(µθ + Wσ 2θ 2) 1 x–µ exp – W ––––– σ ( ( Normal N(µ, σ2) Continuous f(x) = Uniform (Rectangular) on [a, b] Continuous 1 f(x) = ––––– b–a Exponential Continuous f(x) = λe–λx Geometric Discrete P(X = r) = q r – 1p , σ 2π ) ), 2 –∞ < x < ∞ 9 Negative binomial Discrete , a<x<b , 0<p<1 x > 0, λ > 0 r = 1, 2, ... , , q=1–p P(X = r) = r – 1Cn – 1 qr – n pn , r = n, n + 1, ... , 0<p<1 , a+b ––––– 2 1 –– 12 (b – a)2 ebθ – eaθ M(θ) = ––––––––– (b – a)θ 1 –– λ 1 –– λ2 λ M(θ) = ––––– λ–θ 1 –– p q ––2 p pt G(t) = ––––– 1 – qt n –– p nq ––2 p pt G(t) = ––––– 1 – qt q=1–p Ctanujit Classes ( STATISTICS: DISTRIBUTIONS Binomial B(n, p) Discrete ) n Page No. 12 Numerical Solution of Equations f(xn) The Newton-Raphson iteration for solving f(x) = 0 : xn + 1 = xn – –––– f'(xn) Numerical integration The trapezium rule b 1 b–a ydx ≈ –2 h{(y0 + yn) + 2(y1 + y2 + ... + yn – 1)}, where h = ––––– n a ∫ The mid-ordinate rule ∫a ydx ≈ h(y b 1 – 2 h2 f(a + h) = f(a) + hf '(a) + ––– f"(a + ξ), 0 < ξ < h 2! f(x) (x – a)2 = f(a) + (x – a)f '(a) + –––––– f"(a) + error 2! f(x) (x – a)2 = f(a) + (x – a)f '(a) + –––––– f"(η) , a < η < x 2! b–a + y1 1– + ... + yn – 1 1– + yn – 1– ), where h = ––––– n 2 2 2 ∫a ydx ≈ 1–3 h{(y0 + yn) + 4(y1 + y3 + ... + yn – 1) + 2(y2 + y4 + ... + yn– 2)}, b b–a where h = ––––– n The Gaussian 2-point integration rule 10 −h h f ( x )dx ≈ h f + f –h 3 3 Interpolation/finite differences h Numerical solution of differential equations dy For –– = f(x, y): dx Euler’s method : yr + 1 = yr + hf(xr, yr); xr+1 = xr + h Runge-Kutta method (order 2) (modified Euler method) yr + 1 = yr + 1–2 (k1 + k2) where k1 = h f(xr, yr), k2 = h f(xr + h, yr + k1) Runge-Kutta method, order 4: n x – xi Lagrange’s polynomial : Pn(x) = ∑ Lr(x)f(x) where Lr(x) = ∏ –––––– x i=0 r – xi i≠r Newton’s forward difference interpolation formula (x – x0)(x – x1) (x – x0) –––––––––––– f(x) = f(x0) + –––––– ∆f(x ) + ∆2f(x0) + ... 0 2!h2 h Newton’s divided difference interpolation formula yr+1 = yr + 1–6 (k1 + 2k2 + 2k3 + k4), where k1 = hf(xr, yr) k3 = hf(xr + 1–2 h, yr + 1–2 k2) k2 = hf(xr + 1–2 h, yr + 1–2 k1) k4 = hf(xr + h, yr + k3). NUMERICAL ANALYSIS DECISION & DISCRETE MATHEMATICS Simpson’s rule for n even ∫ Taylor polynomials h2 f(a + h) = f(a) + hf '(a) + ––– f"(a) + error 2! Logic gates f(x) = f[x0] + (x – x0]f[x0, x1] + (x – x0) (x – x1)f[x0, x1, x2] + ... Numerical differentiation f(x + h) – 2f(x) + f(x – h) f"(x) ≈ ––––––––––––––––––––– h2 NOT AND OR NAND Ctanujit Classes Page No. 13 Statistics formula sheet This has mean nθ and variance nθ(1 − θ). The Poisson distribution: Summarising data Sample mean: x= This has mean λ and variance λ. n 1X xi . n i=1 Continuous distributions Sample variance: s2x λx exp(−λ) for x = 0, 1, 2, . . . . x! p(x) = n 1 X 1 = (xi − x)2 = n−1 n−1 n X i=1 x2i 2 − nx ! Distribution function: . F (y) = P (X ≤ y) = i=1 Z y f (x) dx. −∞ Sample covariance: g= 1 n−1 n X Density function: (xi −x)(yi −y) = i=1 1 n−1 n X xi yi − nx y ! . d F (x). dx f (x) = i=1 Evaluating probabilities: Sample correlation: g r= . sx sy P (a < X ≤ b) = Z b f (x) dx = F (b) − F (a). a Expected value: Probability E(X) = µ = Addition law: Z ∞ xf (x) dx. −∞ P (A ∪ B) = P (A) + P (B) − P (A ∩ B). Multiplication law: Variance: Var(X) = Z ∞ (x − µ)2 f (x) dx = −∞ P (A ∩ B) = P (A)P (B|A) = P (B)P (A|B). Z ∞ x2 f (x) dx − µ2 . −∞ Hazard function: Partition law: For a partition B1 , B2 , . . . , Bk P (A) = k X P (A ∩ Bi ) = i=1 k X h(t) = P (A|Bi )P (Bi ). i=1 Bayes’ formula: Normal density with mean µ and variance σ 2 : 1 P (A|Bi )P (Bi ) P (A|Bi )P (Bi ) P (Bi |A) = = Pk . P (A) P (A|Bi )P (Bi ) i=1 f (t) . 1 − F (t) f (x) = √ exp 2πσ 2  1 − 2  x−µ σ 2  for x ∈ [−∞, ∞]. Weibull density: f (t) = λκtκ−1 exp(−λtκ ) for t ≥ 0. Discrete distributions Exponential density: Mean value: E(X) = µ = X f (t) = λ exp(−λt) for t ≥ 0. xi p(xi ). This has mean λ−1 and variance λ−2 . xi ∈S Variance: Var(X) = X (xi − µ)2 p(xi ) = xi ∈S X Test for population mean x2i p(xi ) − µ2 . Data: Single sample of measurements x1 , . . . , xn . xi ∈S Hypothesis: H : µ = µ0 . The binomial distribution: p(x) =   n x θ (1 − θ)n−x for x = 0, 1, . . . , n. x Method: √ • Calculate x, s2 , and t = |x − µ0 | n/s. • Obtain critical value from t-tables, df = n − 1. Ctanujit Classes Page No. 14 • Reject H at the 100p% level of significance if |t| > c, where c is the tabulated value corresponding to column p. • Calculate s2 = (n − 1)s2x + (m − 1)s2y /(n + m − 2).  • Look in t-tables, df = n + m − 2, column p. Let the tabulated value be c say. Paired sample t-test Data: Single sample of n measurements x1 , . . . , xn which are the pairwise differences between the two original sets of measurements. • 100(1 − p)% confidence interval for the difference in population means i.e. µx − µy , is (x − y) ± c Hypothesis: H : µ = 0. (r s2  1 1 + n m  ) . Method: √ • Calculate x, s2 and t = x n/s. • Obtain critical value from t-tables, df = n − 1. Regression and correlation • Reject H at the 100p% level of significance if |t| > c, where c is the tabulated value corresponding to column p. The linear regression model: yi = α + βxi + zi . Least squares estimates of α and β: Two sample t-test Data: Two separate samples of measurements x1 , . . . , xn and y1 , . . . , ym . Hypothesis: H : µx = µy . βˆ = Pn xi yi − n x y , (n − 1)s2x i=1 and α ˆ = y − βˆ x. Confidence interval for β Method: • Calculate x, s2x , y, and s2y . • Calculate βˆ as given previously. • Calculate s2ε = s2y − βˆ2 s2x . • Calculate 2 s2 = (n − 1)s2x + (m − 1)sy /(n + m − 2).  x−y . 1 1 s2 + n m • C...
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  • Winter '18
  • manish
  • Prime number, Even and odd functions, Parity, Evenness of zero

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