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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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- Prime number, Even and odd functions, Parity, Evenness of zero
