University of Bath Formula Book

Revised 2005

Contents

1 Algebraic and Trigonometrical Formulae 1

2 Hyperbolic Functions 2

3 Derivatives 3

4 Integrals 4

5 Differentiation under the Integral Sign 5

6 Coordinate Geometry 5

7 Series 6

8 Taylor’s Series for Two Variables 7

9 Numerical Formulae 8

10 Fourier Series 10

11 Fourier Transforms 11

12 Laplace Transforms 13

13 Vector Formulae 18

14 Curvilinear Coordinates 19

15 Index Notation Formulae 20

16 The Normal Distribution Function 21

17 Percentage Points of the Normal (Gaussian) Distribution 21

18 Percentage Points of Student’s t -Distribution 22

19 Percentage Points of the χ2 -distribution 23

20 Percentage Points of the F -Distribution 24

21 Poisson Tables 28

22 Legendre Polynomials 32

23 Orthogonal Polynomials 33

24 Random Numbers 34

25 Wilcoxon Matched-Pairs Test 35

26 Mann-Whitney Test 36

27 Rank Correlation Coefficients (Spearman’s) 38

28 Correlation Coefficients 38

29 Constants for Use in Constructing Quality Control Charts 39

30 Some Common Families of Distributions 40

1 Algebraic and Trigonometrical Formulae

(a± b)2 = a2 ± 2ab+ b2

(a± b)3 = a3 ± 3a2b+ 3ab2 ± b3

a2 − b2 = (a+ b)(a− b)

a3 ± b3 = (a± b)(a2 ∓ ab+ b2)

a2 + b2 has no real factors

(a+ b)n =n∑

r=0

nCr an−rbr

where nCr =n!

r!(n− r)! =n(n− 1) · · · (n− r + 1)

r!. Also written as

(n

r

).

sin(A±B) = sinA cosB ± cosA sinB

cos(A±B) = cosA cosB ∓ sinA sinB

tan(A±B) =tanA± tanB

1∓ tanA tanB

sin 2A = 2 sinA cosA

cos 2A = cos2A− sin2A = 2 cos2A− 1 = 1− 2 sin2A

tan 2A =2 tanA

1− tan2A

sin θ =2t

1 + t2; cos θ =

1− t21 + t2

where t = tan θ2

.

2 sinA cosB = sin(A+B) + sin(A−B)

2 cosA cosB = cos(A+B) + cos(A−B)

2 sinA sinB = cos(A−B)− cos(A+B)

sinA± sinB = 2 sin(A±B

2

)cos(A∓B

2

)

cosA+ cosB = 2 cos(A+B

2

)cos(A−B

2

)

cosA− cosB = −2 sin(A+B

2

)sin(A−B

2

)

Sine Rule:a

sinA=

b

sinB=

c

sinC

Cosine Rule: a2 = b2 + c2 − 2bc cosA

1

2 Hyperbolic Functions

sinhx = 12(ex − e−x)

coshx = 12(ex + e−x)

tanhx =sinhx

coshx

cothx =coshx

sinhx

sechx =1

coshx

cosechx =1

sinhx

cosh2 x− sinh2 x = 1

1− tanh2 x = sech2 x

coth2 x− 1 = cosech2 x

sinh(x± y) = sinh x cosh y ± coshx sinh y

cosh(x± y) = cosh x cosh y ± sinhx sinh y

tanh(x± y) =tanhx± tanh y

1± tanhx tanh y

sinh 2x = 2 sinh x coshx

cosh 2x = cosh2 x+ sinh2 x = 2 cosh2 x− 1 = 1 + 2 sinh2 x

tanh 2x =2 tanhx

1 + tanh2 x

sinh−1 x = ln(x+√x2 + 1)

cosh−1 x = ln(x+√x2 − 1) (x ≥ 1)

tanh−1 x = 12

ln

{1 + x

1− x

}(−1 < x < 1)

2

3 Derivatives

ydy

dx

tanx sec2 x

cotx − cosec2 x

secx secx tanx

cosecx − cosecx cotx

tanhx sech2 x

cothx − cosech2 x

sechx − sechx tanhx

cosechx − cosechx cothx

sin−1 x1√

1− x2

cos−1 x−1√

1− x2

tan−1 x1

1 + x2

sec−1 x1

x√x2 − 1

cosec−1 x−1

x√x2 − 1

cot−1 x−1

1 + x2

sinh−1 x1√

x2 + 1

cosh−1 x1√

x2 − 1

tanh−1 x1

1− x2

sech−1 x−1

x√

1− x2

cosech−1 x−1

x√

1 + x2

coth−1 x−1

x2 − 1

3

4 Integrals

f(x)

∫f(x) dx

tanx ln(secx)

cotx ln(sinx)

secx ln(secx+ tanx) = ln tan(x2

+ π4

)= 1

2ln(

1+sinx1−sinx

)

cosecx − ln(cosecx+ cotx) = ln tan x2

= 12

ln(

1−cosx1+cosx

)

sechx 2 tan−1(ex)

cosechx ln(tanh x

2

)

1

a2 − x21a

tanh−1(xa

)= 1

2aln(a+xa−x

)

1

x2 − a2− 1a

coth−1(xa

)= 1

2aln(x−ax+a

)

1

a2 + x21a

tan−1(xa

)

1√a2 − x2

sin−1(xa

)

1√x2 − a2

cosh−1(xa

)= ln

(x+√x2−a2a

)

1√a2 + x2

sinh−1(xa

)= ln

(x+√a2+x2

a

)

√a2 − x2 x

2

√a2 − x2 + a2

2sin−1

(xa

)

√x2 − a2 x

2

√x2 − a2 − a2

2cosh−1

(xa

)

√a2 + x2 x

2

√a2 + x2 + a2

2sinh−1

(xa

)

∫ π2

0

sinn x dx =

∫ π2

0

cosn x dx =(n− 1)!!

n!!×

π2, n even

1, n odd

∫ π2

0

sinm x cosn x dx =(m− 1)!!(n− 1)!!

(m+ n)!!×

π2, m and n both even

1, otherwise

where p!! = p(p− 2)(p− 4) · · · · ·2 or 1 and 0!! = 1 .

∫eax sin bx dx =

eax

a2 + b2(a sin bx− b cos bx)

∫eax cos bx dx =

eax

a2 + b2(b sin bx+ a cos bx)

4

5 Differentiation under the Integral Sign

d

dx

∫ v(x)

u(x)

f(x, t) dt =

∫ v(x)

u(x)

∂

∂x{f(x, t)} dt +

dv

dxf (x, v(x)) − du

dxf (x, u(x)) .

6 Coordinate Geometry (Two Dimensions)

Straight line: y = mx+ C , gradient m , intercept C on y axis.

Data

Semi

latus

Conic Cartesian Eccentricity rectum

Section Equation (e) Foci (`)

Circle (x− a)2 + (y − b)2 e = 0 (0, 0) R Centre (a, b)

= R2 radius R

Ellipsex2

a2+y2

b2= 1 0 < e < 1 (±ae, 0)

b2

ab2 = a2(1− e2)

(a > b)

Hyperbolax2

a2− y2

b2= 1 e > 1 (±ae, 0)

b2

ab2 = a2(e2 − 1)

asymptotes

y = ± bax

Rect. xy = c2 (constant) e =√

2 (±c√

2,±c√

2) c√

2 asymptotes

Hyperbola x = 0, y = 0

Parabola y2 = 4ax e = 1 (a, 0) 2a Vertex (0, 0)

Polar equation for all conic sections ` = r(1 + e cos θ)

5

7 Series

a = (a+ d) + (a+ 2d) + · · ·+ (a+ |n− 1|d) = n2(2a+ |n− 1|d)

1 + r + r2 + · · ·+ rn =1− rn+1

1− r

1 + 2 + 3 + · · ·+ n = 12n(n+ 1)

12 + 22 + 32 + · · ·+ n2 = 16n(n+ 1)(2n+ 1)

13 + 23 + 33 + · · ·+ n3 = 14n2(n+ 1)2

(1 + x)n = 1 + nx + n(n−1)1.2

x2 + n(n−1)(n−2)1.2.3

x3 + · · · |x| < 1

(1 + x)−1 = 1 − x + x2 − x3 + · · · |x| < 1

ex = 1 + x + x2

2!+ x3

3!+ · · · All x

ln(1 + x) = x − x2

2+ x3

3− x4

4+ · · · |x| < 1

sinx = x − x3

3!+ x5

5!− x7

7!+ · · · All x

cosx = 1 − x2

2!+ x4

4!− x6

6!+ · · · All x

tanx = x + x3

3+ 2x5

15− 17

315x7 + · · · |x| < π

2

sin−1 x = x + 12x3

3+ 1.3

2.4x5

5+ 1.3.5

2.4.6x7

7+ · · · |x| < 1

cos−1 x = π2− sin−1 x

tan−1 x = x − x3

3+ x5

5− x7

7+ · · · |x| < 1

sinhx = x + x3

3!+ x5

5!+ x7

7!+ · · · All x

coshx = 1 + x2

2!+ x4

4!+ x6

6!+ · · · All x

tanhx = x − x3

3+ 2x5

15− 17

315x7 + · · · |x| < π

2

sinh−1 x = x − 12x3

3+ 1.3

2.4x5

5− 1.3.5

2.4.6x7

7+ · · · |x| < 1

cosh−1 x = ln 2x − 12

12x2− 1.3

2.41

4x4− 1.3.5

2.4.61

6x6− · · · x > 1

tanh−1 x = x + x3

3+ x5

5+ x7

7+ · · · |x| < 1

6

Maclaurin’s Series:

f(x) = f(0) + xf ′(0) +x2

2!f ′′(0) + · · · +

xnf (n)(0)

n!+ Rn+1

where Rn+1 = xn+1f(n+1)(θx)

(n+ 1)!(0 < θ < 1)

Taylor’s Series

f(x) = f(a) + hf ′(a) +h2

2!f ′′(a) + · · · +

hnf (n)(a)

n!+ Rn+1

where h = x− a

and Rn+1 =1

n!

∫ x

a

(x− s)nf (n+1)(s) ds

= hn+1f(n+1)(a+ θh)

(n+ 1)!(0 < θ < 1)

8 Taylor’s Series for Two Variables

f(x, y) = f(a, b) +

(h∂

∂x+ k

∂

∂y

)f(a, b)

+1

2!

(h2 ∂

2

∂x2+ 2hk

∂2

∂x∂y+ k2 ∂

2

∂y2

)f(a, b) + · · ·

+1

n!

(h∂

∂x+ k

∂

∂y

)nf(a, b) + · · ·

where h = x− a, k = y − b.

7

9 Numerical Formulae

Trapezium rule

∫ a+h

a

f(x) dx =h

2(f0 + f1) + E

where E = − 1

12h3f ′′(X), a < X < a+ h

b = a+ nh,

∫ b

a

f(x) dx = h

(1

2f0 + f1 + f2 + · · ·+ fn−1 +

1

2fn

)+ E

where E = − 1

12h2(b− a)× (Average value of f ′′)

Simpson’s rule

∫ a+2h

a

f(x) dx =h

3(f0 + 4f1 + f2) + E

where E = − 1

90h5f (4)(X), a < X < a+ 2h

b = a+ 2nh,

∫ b

a

f(x) dx =h

3(f0 + 4f1 + 2f2 + 4f3 + 2f4 + · · ·+ 2f2n−2 + 4f2n−1 + f2n) + E

where E = − 1

180h4(b− a)× (Average value of f (4))

Newton’s formula for roots of equations f(x) = 0

xn+1 = xn −f(xn)

f ′(xn)

Step-by-step integration of differential equations (Modified Euler)

y(P )1 = y0 + hy′0

y(P )n+1 = yn−1 + 2hy′n Error 1

3h3y′′′n + higher order terms

y(C)n+1 = yn + h

2(y′n+1 + y′n) Error − 1

12h3y′′′n + higher order terms

8

The Lagrange interpolation formula

If f ∈ C(n+1)[a, b] and a ≤ x0 < x1 < · · · < xn ≤ b then for x ∈ [a, b]

f(x) =n∑

j=0

`j,n(x)f(xj) +Pn(x)

(n+ 1)!fn+1(ξ)

where

`j,n(x) =n∏

k=0k 6=j

[x− xkxj − xk

]=

Pn(x)

(x− xj)P ′n(xj)

Pn(x) =n∏

k=0

[x− xk]

and a ≤ ξ ≤ b.

The (modified) Hermite interpolation formula

If f ∈ C(n+r+2)[a, b] and a ≤ x0 < x1 < · · · < xn ≤ b then for x ∈ [a, b]

f(x) = y(x) + E(x)

where

y(x) =n∑

j=0

hj(x)f(xj) +r∑

j=0

hj(x)f ′(xj), r ≤ n,

with

hj(x) =

[1− (x− xj){`′j,n(xj) + `′j,r(xj)}]`j,n(x)`j,r(x), j = 0, 1, · · · , r;

`j,n(x)Pr(x)/Pr(xj), j = r + 1, r + 2, · · · , n.

hj(x) = (x− xj)`j,n(x)`j,r(x), j = 0, 1, · · · , r

and

E(x) =Pn(x)Pr(x)

(n+ r + 2)!f (n+r+2)(ξ)

and a ≤ ξ ≤ b .

9

10 Fourier Series

(a) f(t) periodic, period T, fundamental frequency ω : ωT = 2π

(i) real form:

f(t) = c0 +∞∑

n=1

cn sin(nωt+ αn) =a0

2+∞∑

n=1

(an cosnωt+ bn sinnωt)

an =2

T

∫ θ+T

θ

f(t) cosnωt dt = twice mean value of f(t) cosnωt

over a period. (θ arbitrary)

bn =2

T

∫ θ+T

θ

f(t) sinnωt dt = twice mean value of f(t) sinnωt

over a period. (θ arbitrary)

(ii) complex form:

f(t) =∞∑

n=−∞

Cneinωt

Cn =1

T

∫ θ+T

θ

f(t)e−inωt dt = mean value of f(t)e−inωt over a period

(θ arbitrary)

(b) g(x) defined for 0 < x < ` .

g(x) =∞∑

1

bn sin(nπx

`

)where bn =

2

`

∫ `

0

g(x) sin(nπx

`

)dx

g(x) =a0

2+∞∑

1

an cos(nπx

`

)where an =

2

`

∫ `

0

g(x) cos(nπx

`

)dx

10

11 Fourier Transforms

If∫ ∞

−∞|g(t)| dt < ∞ then F [g(t)] =

∫ ∞

−∞g(t)e−jωt dt = G(ω)

and F−1[G(ω)] =1

2π

∫ ∞

−∞G(ω)ejωt dω = g(t)

If g(t) = 0 for t < 0 and g(s) has no poles in Re(s) 1 0 then

G(ω) = F [g(t)] = L[g(t)]s = jω = g(jω)

g(t) G(ω) = F [g(t)]

Even Function g(t) = g(−t) G(ω) = G(−ω) = 2

∫ ∞

0

g(t) cos ωt dt

Odd Function g(t) = −g(−t) G(ω) = −G(−ω) = −2j

∫ ∞

0

g(t) sin ωt dt

Symmetry G(t) 2πg(−ω)

Reflection g(−t) G(−ω)

Conjugate g∗(t) G∗(−ω)

Scale change g(tT

), (T > 0) TG(ωT )

Derivativedg(t)

dtj ω G(ω)

tg(t) jdG(ω)

dω

Time Shift g(t + τ) ejωτG(ω)

Frequency Shift g(t)ejω0 t G(ω − ω0)

Convolution (f ∗ g)(t) F (ω)G(ω)

Frequency f(t)g(t) 12π

(F ∗G)(ω)

convolution

where (f ∗ g) (x)4=

∫ ∞

−∞f(y)g(x − y) dy =

∫ ∞

−∞f(x − y)g(y) dy

Parseval’s theorem∫ ∞

−∞f(t)g(t) dt =

1

2π

∫ ∞

−∞F (ω)G(−ω) dω

and∫ ∞

−∞|g(t)|2 dt =

1

2π

∫ ∞

−∞|G(ω)|2 dω

11

Transform Pairs

g(t) G(ω) = F g(t)

δ (t) 1

ejω0t 2πδ(ω − ω0)

sgn (t)4=

1, t > 0

−1, t < 02/jω

H(t) πδ(ω) +1

jω

rect

(t

τ

)=

H(t+ τ2)−H(t− τ

2)

1, |t| < τ/2

0, |t| > τ/2

sinωτ/2

ω/2

sinω0t

πtH(ω + ω0)−H(ω − ω0)

e−at2, (a > 0)

√π

ae−ω

2/4a

1

a2 + t2, (a > 0)

π

ae−a|ω|

e−a|t|, (a > 0)2a

a2 + ω2

12

12 Laplace Transforms

Lf(t) = f(s) =

∫ ∞

0

f(t)e−st dt

Operational Form

f(t)H(t) = f(s)δ(t)

1

s≡∫ t

0

( ) dt; s ≡ d

dt

Functional Relationships

f(t) f(s)

f ′(t) sf(s)− f(0)

f ′′(t) s2f(s)− [sf(0) + f ′(0)]

f (n)(t) snf(s)−[sn−1f(0) + sn−2f ′(0) + · · ·+ f (n−1)(0)

]

∫ t

0

f(t)dt 1sf(s)

Damping e−ktf(t) f(s + k)

Delay f(t− T )H(t− T ) e−sTf(s)

Scale change f(kt) 1kf (s/k)

Periodic, period T f(t) f(s) =1

1− e−sT∫ T

0

f(t)e−stdt

Convolution

f(t) ∗ g(t) =

∫ t

0

f(r)g(t− r)dr

=

∫ t

0

f(t− r)g(r)dr

f(s)g(s)

∫ t

0

· · ·∫ t

0

f(t)(dt)n 1snf(s)

tnf(t) (−1)ndn

dsn(f(s)

)

1tnf(t)

∫ ∞

s

· · ·∫ ∞

s

f(s)(ds)n

13

A second independent variable

f(t, x) f(s, x) =

∫ ∞

0

f(t, x)e−st dt

∂

∂tf(t, x) sf(s, x)− f(0, x)

∂2

∂t2f(t, x) s2f(s, x)−

[sf(0, x) +

∂f

∂t(0, x)

]

∂

∂xf(t, x)

∂

∂xf(s, x)

∫ t

t=0

f(t, x) dt1

sf(s, x)

∫ b

x=a

f(t, x) dx

∫ b

x=a

f(s, x) dx

Limiting Values

limt→+ 0

f(t) = lims→+∞

sf(s)

limt→+∞

f(t) = lims→+0

sf(s)

∫ ∞

0

f(t) dt = lims→+0

f(s)

(If limits and integral exist)

Inversion Integral

f(t) = L−1f(s) =1

2πj

∫ γ+j∞

γ−j∞f(s)est ds =

∑Resf(s)est

if f(s) analytic except for poles in LH1

2plane

Partial Fractions

Simple P/Q =P

(s+ s1)(s+ s2) · · · =A1

(s+ s1)+

A2

(s+ s2)+ · · ·+ Ar

(s+ sr)+ · · ·

Ar = [(s+ sr)P/Q]s=−sr = [P/Q′]s=−sr (“Cover up”)

Double P/Q =P

(s+ s1)2(s+ s2) · · · =A1

1

(s+ s1)+

A1

(s+ s1)2+

A2

(s+ s2)+ · · ·

A1 = [(s+ s1)2P/Q]−s1 A11 =

[d

ds(s+ s1)2P/Q

]

−s1

14

Laplace Transforms Of Simple Functions

f(t) f(s)

δ(t) = u0(t) 1

H(t) = u−1(t) = U(t)1

s

tU(t) = u−2(t)1

s2

tn

Γ(n+1)sn+1 for n > −1

n!sn+1 for n positive integer

e−kt1

s+ k

te−kt1

(s+ k)2

sinωtω

s2 + ω2

cosωts

s2 + ω2

e−kt sinωtω

(s+ k)2 + ω2

e−kt cosωts+ k

(s+ k)2 + ω2

Square

Wave

Period 2T

f(t) =

1 0 < t < T

−1 T < t < 2T

1

s

1− e−sT1 + e−sT

=1

stanh sT/2

Triangular

Period 2Tf(t) =

t/T 0 < t < T

−(t−2T )T

T < t < 2T

1

Ts2

1− e−sT1 + e−sT

=1

Ts2tanh sT/2

Saw Tooth

Period Tf(t) = t/T 0 < t < T

1

Ts2− e−sT

s(1−e−sT )

Rectified

Wavesf(t) = | sinωt| ω

s2 + ω2

1 + e−sπ/ω

1− e−sπ/ω =ω

s2 + ω2coth

sπ

2ω

Angular

Frequency

ω

f(t) =

sinωt 0 < t < πω

0 πω< t < 2π

ω

ω

s2 + ω2

1

1− e−sπ/ω

15

Inverse Laplace Transforms

f(s) f(t)1

(s+ a)(s+ b)

1

b− a(e−at − e−bt

)

1

(s+ a)2t e−at

s

s2 + ω2cosωt

1

s2 + ω2

1

ωsinωt

1

s2(s+ a)

1

a2(at− 1 + e−at)

1

s(s2 + ω2)

1

ω2(1− cosωt)

1

s2(s2 + ω2)

1

ω3(ωt− sinωt)

s

(s2 + ω2)2

1

2ωt sinωt

1

(s2 + ω2)2

1

2ω3(sinωt− ωt cosωt)

1

s(s2 + ω2)2

1

ω4

(1− cosωt− ωt

2sinωt

)

s

(s2 + a2)(s2 + ω2)

1

a2 − ω2(cosωt− cos at)

1

(s2 + a2)(s2 + ω2)

1

aω(a2 − ω2)(a sinωt− ω sin at)

1

s(s2 + a2)(s2 + ω2)

1

a2ω2

{1− 1

a2 − ω2(a2 cosωt− ω2 cos at)

}

In the following formulae w2 = c2 − k2. If ω2 < 0, refer to first expression at top of page.s

s2 + 2ks+ c2e−kt

(cosωt− k

ωsinωt

)

1

s2 + 2ks+ c2

1

ωe−kt sinωt

1

s(s2 + 2ks+ c2)

1

c2

{1− e−kt

(cosωt+

k

ωsinωt

)}

1

s2(s2 + 2ks+ c2)

1

c4

{c2t− 2k + e−kt

(2k cosωt+

k2 − ω2

ωsinωt

)}

s

(s+ a)(s2 + 2ks+ c2)

1

A

{−ae−at + e−kt

(a cosωt+

c2 − akω

sinωt

)}

1

(s+ a)(s2 + 2ks+ c2)

1

A

{e−at − e−kt

(cosωt+

k − aω

sinωt

)}

1

s(s+ a)(s2 + 2ks+ c2)

1

ac2+

1

A

{−e−at

a− e−kt

(B cosωt+

kB + 1

ωsinωt

)}

where A = (a− k)2 + ω2 and B = (a− 2k)/c2

16

Laplace Transforms Of Special Functions

f(s) f(t)

e−k√s k

2√πt3

exp

(−k2

4t

)(k > 0)

1

se−k√s erfc

(k

2√t

)(k ≥ 0)

1√se−k√s 1√

πtexp

(−k2

4t

)(k ≥ 0)

1√s3e−k√s 2

√t

πexp

(−k2

4t

)− k erfc

(k

2√t

)(k ≥ 0)

e−b√s+a2

s

1

2

[e−ab erfc

(b− 2at

2√t

)+ eab erfc

(b+ 2at

2√t

)]

e−k√s

b+√s

1√πt

exp

(−k2

4t

)− bekb+b2t erfc

{k + 2bt

2√t

}(k 1 0, b 1 0)

e−k√s

s+ b√s

ekb+b2t erfc

{k + 2bt

2√t

}(k 1 0, b 1 0)

e−k√s

s(b+√s)

1

berfc

(k

2√t

)− 1

bekb+b

2t erfc

{k + 2bt

2√t

}(k 1 0, b 1 0)

K0(a√s)

1

2texp

(−a2

4t

)

1√s2 + a2

J0(at)

{√s2 + a2 − s

}n

an√s2 + a2

Jn(at), (n > −1)

1√s2 − a2

I0(at)

{s−√s2 − a2

}n

an√s2 − a2

In(at), (n > −1)

bne−b/s

sn+1(bt)n/2Jn(2

√bt) (n > −1)

17

13 Vector Formulae

Scalar Product a.b = ab cos θ = a1b1 + a2b2 + a3b3

Vector Product a ∧ b = ab sin θ n̂ =

∣∣∣∣∣∣∣∣

i j k

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣∣∣Triple Products [a,b, c] = (a ∧ b).c

= a.(b ∧ c) =

∣∣∣∣∣∣∣∣

a1 a2 a3

b1 b2 b3

c1 c2 c3

∣∣∣∣∣∣∣∣

a ∧ (b ∧ c) = (a.c)b− (a.b)c

Vector Calculus O ≡(i∂

∂x+ j

∂

∂y+ k

∂

∂z

)

grad φ ≡ Oφ, div A ≡ O.A, curl ≡ O ∧A

O(φψ) = φOψ + ψOφ

O.(φA) = φO.A + A.Oφ

O ∧ (φA) = φO ∧A + Oφ ∧A

O.(A ∧B) = B.O ∧A−A.O ∧B

O ∧ (A ∧B) = (O.B)A− (O.A)B + (B.O)A− (A.O)B

O(A.B) = A ∧ (O ∧B) + B ∧ (O ∧A) + (A.O)B + (B.O)A

O.Oφ ≡ O2φ

O.(O ∧A) = 0

O ∧ (Oφ) = 0

O2A = O(O.A)− O ∧ (O ∧A)

A ∧ (O ∧A) = O(12A2)−AOA

Integral Theorems

Divergence Theorem:∫

S

A.dS =

∫

V

O.A dV

∫

S

φ dS =

∫

V

Oφ dV

Stokes Theorem:∫

S

(OA).dS =

∮

C

A.dr

∫

S

dS ∧ Oφ =

∮

C

φ dr

Green’s Theorems:∫

V

(Oφ).(Oψ)dV +

∫

V

φO2ψ dV =

∫

S

φOψ.dS =

∫

S

φ∂ψ

∂ndS

∫

V

(ψO2φ− φO2ψ) dV =

∫

S

(ψOφ− φOψ).dS

18

14 Curvilinear Coordinates

General orthogonal co-ordinates ( u1, u2, u3 )

Oφ =

(1

h1

∂φ

∂u1

,1

h2

∂φ

∂u2

,1

h3

∂φ

∂u3

)

div A =1

h1h2h3

{∂

∂u1

(h2h3A1) +∂

∂u2

(h3h1A2) +∂

∂u3

(h1h2A3)

}

curl A =1

h1h2h3

∣∣∣∣∣∣∣∣∣∣∣∣

h1e1 h2e2 h3e3

∂

∂u1

∂

∂u2

∂

∂u3

h1A1 h2A2 h3A3

∣∣∣∣∣∣∣∣∣∣∣∣

O2φ =1

h1h2h3

{∂

∂u1

(h2h3

h1

∂φ

∂u1

)+

∂

∂u2

(h3h1

h2

∂φ

∂u2

)+

∂

∂u3

(h1h2

h3

∂φ

∂u3

)}

Line element: δs1 = h1δu1;

δs2 = h2δu2;

δs3 = h3δu3;

Surface element: δS1 = h2h3δu2δu3

δS2 = h3h1δu3δu1

δS3 = h1h2δu1δu2

Volume element: δV = h1h2h3δu1δu2δu3

Co-ordinates u1 u2 u3 h1 h2 h3 Cartesian/polar relation

Rectangular x y z 1 1 1 x y z

Cylindrical ρ φ z 1 ρ 1 ρ cosφ ρ sinφ z

Spherical r θ φ 1 r r sin θ r sin θ cosφ r sin θ sinφ r cos θ

Form for O2V (scalars only);

Cylindrical Polars:1

ρ

∂

∂ρ

(ρ∂V

∂ρ

)+

1

ρ2

∂2V

∂φ2+∂2V

∂z2

Spherical Polars:1

r2

∂

∂r

(r2∂V

∂r

)+

1

r2 sin θ

∂

∂θ

(sin θ

∂V

∂θ

)+

1

r2 sin2 θ

∂2V

∂φ2

19

15 Index Notation Formulae

δij =

1 i = j

0 i 6= j

εijk =

+1 (ijk) cyclic in (123)

−1 (ijk) anticyclic in (123)

0 otherwise

εkijεkpq ≡ εijkεpqk = δipδjq − δiqδjp

(a× b) = εijkajbk

Divergence Theorem

∫

V

∂φ

∂xidV =

∫

S

φdSi, φ is scalar, vector or tensor

20

16 The Normal Distribution Function Φ(z)

THE NORMAL DISTRIBUTION FUNCTION Φ(z)

z

Φ(z) = P (Z < z) = 1√2π

� z

−∞ e−t2/2 dt

z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.53590.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.57530.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.61410.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.65170.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.68790.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.72240.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.75490.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.78520.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.83891.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.86211.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.88301.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.90151.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.91771.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.93191.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.94411.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.96331.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.97061.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.97672.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.98172.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.98572.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.98902.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.99162.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.99362.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.99522.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.99642.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.99742.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.99812.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.99863.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.99903.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993

PERCENTAGE POINTS OF THE NORMAL DISTRIBUTION

The value is that at which the upper tail probability equals the product of the row and columnlabels, rounded up in the 3rd D.P.

10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8 10−9 10−10

5.0 0.000 1.645 2.576 3.291 3.891 4.417 4.892 5.327 5.731 6.1092.5 0.674 1.960 2.807 3.481 4.056 4.565 5.026 5.451 5.847 6.2191.0 1.282 2.326 3.090 3.719 4.265 4.753 5.199 5.612 5.998 6.361

Φ(z) = P (Z < z) =1√2π

∫ z

−∞e−t

2/2 dt

z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359

0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753

0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141

0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517

0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879

0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224

0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549

0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852

0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133

0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389

1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621

1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830

1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015

1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177

1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319

1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441

1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545

1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633

1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706

1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767

2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817

2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857

2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890

2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916

2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936

2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952

2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964

2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974

2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981

2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986

3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990

3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993

17 Percentage Points of the Normal Distribution

The value is that at which the upper tail probability equals the product of the row and columnlabels, rounded up in the 3rd D.P.

10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8 10−9 10−10

5.0 0.000 1.645 2.576 3.291 3.891 4.417 4.892 5.327 5.731 6.109

2.5 0.674 1.960 2.807 3.481 4.056 4.565 5.026 5.451 5.847 6.219

1.0 1.282 2.326 3.090 3.719 4.265 4.753 5.199 5.612 5.998 6.361

21

18 Percentage Points of Student’s t -Distribution

PERCENTAGE POINTS OF STUDENT’S t-DISTRIBUTION

t!,"

"

The value given is tν,α where P (tν > tν,α) = αfor Student’s t-distribution on ν degrees of freedom.Note that P (|tν | > tν,α/2) = α.

Two-tailed Probabilitiesα/2 0.5 0.1 0.05 0.02 0.01

One-tailed Probabilitiesα 0.25 0.05 0.025 0.01 0.005ν1 1.000 6.314 12.706 31.821 63.6572 0.816 2.920 4.303 6.965 9.9253 0.765 2.353 3.182 4.541 5.8414 0.741 2.132 2.776 3.747 4.6045 0.727 2.015 2.571 3.365 4.0326 0.718 1.943 2.447 3.143 3.7077 0.711 1.895 2.365 2.998 3.4998 0.706 1.860 2.306 2.896 3.3559 0.703 1.833 2.262 2.821 3.250

10 0.700 1.812 2.228 2.764 3.16911 0.697 1.796 2.201 2.718 3.10612 0.695 1.782 2.179 2.681 3.05513 0.694 1.771 2.160 2.650 3.01214 0.692 1.761 2.145 2.624 2.97715 0.691 1.753 2.131 2.602 2.94716 0.690 1.746 2.120 2.583 2.92117 0.689 1.740 2.110 2.567 2.89818 0.688 1.734 2.101 2.552 2.87819 0.688 1.729 2.093 2.539 2.86120 0.687 1.725 2.086 2.528 2.84521 0.686 1.721 2.080 2.518 2.83122 0.686 1.717 2.074 2.508 2.81923 0.685 1.714 2.069 2.500 2.80724 0.685 1.711 2.064 2.492 2.79725 0.684 1.708 2.060 2.485 2.78726 0.684 1.706 2.056 2.479 2.77927 0.684 1.703 2.052 2.473 2.77128 0.683 1.701 2.048 2.467 2.76329 0.683 1.699 2.045 2.462 2.75630 0.683 1.697 2.042 2.457 2.75035 0.682 1.690 2.030 2.438 2.72440 0.681 1.684 2.021 2.423 2.70445 0.680 1.679 2.014 2.412 2.69050 0.679 1.676 2.009 2.403 2.67860 0.679 1.671 2.000 2.390 2.660∞ 0.674 1.645 1.960 2.326 2.576

The value given is tν,α where P (tν > tν,α) = α

for Student’s t-distribution on ν degrees of freedom.Note that P (|tν | > tν,α/2) = α .

Two-tailed probabilitiesα/2 0.5 0.1 0.05 0.02 0.01

One-tailed probabilitiesα 0.25 0.05 0.025 0.01 0.005ν

1 1.000 6.314 12.706 31.821 63.6572 0.816 2.920 4.303 6.965 9.9253 0.765 2.353 3.182 4.541 5.8414 0.741 2.132 2.776 3.747 4.6045 0.727 2.015 2.571 3.365 4.0326 0.718 1.943 2.447 3.143 3.7077 0.711 1.895 2.365 2.998 3.4998 0.706 1.860 2.306 2.896 3.3559 0.703 1.833 2.262 2.821 3.250

10 0.700 1.812 2.228 2.764 3.16911 0.697 1.796 2.201 2.718 3.10612 0.695 1.782 2.179 2.681 3.05513 0.694 1.771 2.160 2.650 3.01214 0.692 1.761 2.145 2.624 2.97715 0.691 1.753 2.131 2.602 2.94716 0.690 1.746 2.120 2.583 2.92117 0.689 1.740 2.110 2.567 2.89818 0.688 1.734 2.101 2.552 2.87819 0.688 1.729 2.093 2.539 2.86120 0.687 1.725 2.086 2.528 2.84521 0.686 1.721 2.080 2.518 2.83122 0.686 1.717 2.074 2.508 2.81923 0.685 1.714 2.069 2.500 2.80724 0.685 1.711 2.064 2.492 2.79725 0.684 1.708 2.060 2.485 2.78726 0.684 1.706 2.056 2.479 2.77927 0.684 1.703 2.052 2.473 2.77128 0.683 1.701 2.048 2.467 2.76329 0.683 1.699 2.045 2.462 2.75630 0.683 1.697 2.042 2.457 2.75035 0.682 1.690 2.030 2.438 2.72440 0.681 1.684 2.021 2.423 2.70445 0.680 1.679 2.014 2.412 2.69050 0.679 1.676 2.009 2.403 2.67860 0.679 1.671 2.000 2.390 2.660∞ 0.674 1.645 1.960 2.326 2.576

22

19 Percentage Points of the χ2 -distribution

PERCENTAGE POINTS OF THE χ2-DISTRIBUTION

!2",#

#

The value given is χ2ν,α where P (χ2

ν > χ2ν,α) = α for

the χ2 distribution on ν degrees of freedom.

α 0.99 0.975 0.95 0.5 0.1 0.05 0.025 0.01ν1 0.000157 0.000982 0.00393 0.455 2.706 3.841 5.024 6.6352 0.0201 0.0506 0.103 1.386 4.605 5.991 7.378 9.2103 0.115 0.216 0.352 2.366 6.251 7.815 9.348 11.3454 0.297 0.484 0.711 3.357 7.779 9.488 11.143 13.2775 0.554 0.831 1.145 4.351 9.236 11.070 12.833 15.0866 0.872 1.237 1.635 5.348 10.645 12.592 14.449 16.8127 1.239 1.690 2.167 6.346 12.017 14.067 16.013 18.4758 1.646 2.180 2.733 7.344 13.362 15.507 17.535 20.0909 2.088 2.700 3.325 8.343 14.684 16.919 19.023 21.666

10 2.558 3.247 3.940 9.342 15.987 18.307 20.483 23.20911 3.053 3.816 4.575 10.341 17.275 19.675 21.920 24.72512 3.571 4.404 5.226 11.340 18.549 21.026 23.337 26.21713 4.107 5.009 5.892 12.340 19.812 22.362 24.736 27.68814 4.660 5.629 6.571 13.339 21.064 23.685 26.119 29.14115 5.229 6.262 7.261 14.339 22.307 24.996 27.488 30.57816 5.812 6.908 7.962 15.338 23.542 26.296 28.845 32.00017 6.408 7.564 8.672 16.338 24.769 27.587 30.191 33.40918 7.015 8.231 9.390 17.338 25.989 28.869 31.526 34.80519 7.633 8.907 10.117 18.338 27.204 30.144 32.852 36.19120 8.260 9.591 10.851 19.337 28.412 31.410 34.170 37.56621 8.897 10.283 11.591 20.337 29.615 32.671 35.479 38.93222 9.542 10.982 12.338 21.337 30.813 33.924 36.781 40.28923 10.196 11.689 13.091 22.337 32.007 35.172 38.076 41.63824 10.856 12.401 13.848 23.337 33.196 36.415 39.364 42.98025 11.524 13.120 14.611 24.337 34.382 37.652 40.646 44.31426 12.198 13.844 15.379 25.336 35.563 38.885 41.923 45.64227 12.879 14.573 16.151 26.336 36.741 40.113 43.195 46.96328 13.565 15.308 16.928 27.336 37.916 41.337 44.461 48.27829 14.256 16.047 17.708 28.336 39.087 42.557 45.722 49.58830 14.953 16.791 18.493 29.336 40.256 43.773 46.979 50.89240 22.164 24.433 26.509 39.335 51.805 55.758 59.342 63.69150 29.707 32.357 34.764 49.335 63.167 67.505 71.420 76.15460 37.485 40.482 43.188 59.335 74.397 79.082 83.298 88.37980 53.540 57.153 60.391 79.334 96.578 101.879 106.629 112.329

100 70.065 74.222 77.929 99.334 118.498 124.342 129.561 135.807

For ν > 100,�

2χ2ν −

√2ν − 1 is approximately distributed as a standard normal.

The value given is χ2ν,α where P (χ2

ν > χ2ν,α) = α for

the χ2 distribution on ν degrees of freedom.

α 0.99 0.975 0.95 0.5 0.1 0.05 0.025 0.01ν

1 0.000157 0.000982 0.00393 0.455 2.706 3.841 5.024 6.6352 0.0201 0.0506 0.103 1.386 4.605 5.991 7.378 9.2103 0.115 0.216 0.352 2.366 6.251 7.815 9.348 11.3454 0.297 0.484 0.711 3.357 7.779 9.488 11.143 13.2775 0.554 0.831 1.145 4.351 9.236 11.070 12.833 15.0866 0.872 1.237 1.635 5.348 10.645 12.592 14.449 16.8127 1.239 1.690 2.167 6.346 12.017 14.067 16.013 18.4758 1.646 2.180 2.733 7.344 13.362 15.507 17.535 20.0909 2.088 2.700 3.325 8.343 14.684 16.919 19.023 21.666

10 2.558 3.247 3.940 9.342 15.987 18.307 20.483 23.20911 3.053 3.816 4.575 10.341 17.275 19.675 21.920 24.72512 3.571 4.404 5.226 11.340 18.549 21.026 23.337 26.21713 4.107 5.009 5.892 12.340 19.812 22.362 24.736 27.68814 4.660 5.629 6.571 13.339 21.064 23.685 26.119 29.14115 5.229 6.262 7.261 14.339 22.307 24.996 27.488 30.57816 5.812 6.908 7.962 15.338 23.542 26.296 28.845 32.00017 6.408 7.564 8.672 16.338 24.769 27.587 30.191 33.40918 7.015 8.231 9.390 17.338 25.989 28.869 31.526 34.80519 7.633 8.907 10.117 18.338 27.204 30.144 32.852 36.19120 8.260 9.591 10.851 19.337 28.412 31.410 34.170 37.56621 8.897 10.283 11.591 20.337 29.615 32.671 35.479 38.93222 9.542 10.982 12.338 21.337 30.813 33.924 36.781 40.28923 10.196 11.689 13.091 22.337 32.007 35.172 38.076 41.63824 10.856 12.401 13.848 23.337 33.196 36.415 39.364 42.98025 11.524 13.120 14.611 24.337 34.382 37.652 40.646 44.31426 12.198 13.844 15.379 25.336 35.563 38.885 41.923 45.64227 12.879 14.573 16.151 26.336 36.741 40.113 43.195 46.96328 13.565 15.308 16.928 27.336 37.916 41.337 44.461 48.27829 14.256 16.047 17.708 28.336 39.087 42.557 45.722 49.58830 14.953 16.791 18.493 29.336 40.256 43.773 46.979 50.89240 22.164 24.433 26.509 39.335 51.805 55.758 59.342 63.69150 29.707 32.357 34.764 49.335 63.167 67.505 71.420 76.15460 37.485 40.482 43.188 59.335 74.397 79.082 83.298 88.37980 53.540 57.153 60.391 79.334 96.578 101.879 106.629 112.329

100 70.065 74.222 77.929 99.334 118.498 124.342 129.561 135.807

For ν > 100 ,√

2χ2ν −√

2ν − 1 is approximately distributed as a standard normal.

23

20 Percentage Points of the F -DistributionPERCENTAGE POINTS OF THE F-DISTRIBUTION

F!1,!2,"

"

The value given is Fν1,ν2,α whereP (Fν1,ν2

> Fν1,ν2,α) = α for the F-distribution withdegrees of freedom ν1 (numerator) and ν2 (denomi-nator).

The value given is Fν1,ν2,α where P (Fν1,ν2 > Fν1,ν2,α) = α

for the F-Distribution with degrees of freedom ν1 (numer-ator) and ν2 (denominator).

Upp

er5%

poin

tsν1

12

34

56

78

910

1215

2024

3040

6012

0∞

ν2 1

161.

419

9.5

215.

722

4.6

230.

223

4.0

236.

823

8.9

240.

524

1.9

243.

924

5.9

248.

024

9.1

250.

125

1.1

252.

225

3.3

254.

32

18.5

119

.00

19.1

619

.25

19.3

019

.33

19.3

519

.37

19.3

819

.40

19.4

119

.43

19.4

519

.45

19.4

619

.47

19.4

819

.49

19.5

03

10.1

39.

559.

289.

129.

018.

948.

898.

858.

818.

798.

748.

708.

668.

648.

628.

598.

578.

558.

534

7.71

6.94

6.59

6.39

6.26

6.16

6.09

6.04

6.00

5.96

5.91

5.86

5.80

5.77

5.75

5.72

5.69

5.66

5.63

56.

615.

795.

415.

195.

054.

954.

884.

824.

774.

744.

684.

624.

564.

534.

504.

464.

434.

404.

366

5.99

5.14

4.76

4.53

4.39

4.28

4.21

4.15

4.10

4.06

4.00

3.94

3.87

3.84

3.81

3.77

3.74

3.70

3.67

75.

594.

744.

354.

123.

973.

873.

793.

733.

683.

643.

573.

513.

443.

413.

383.

343.

303.

273.

238

5.32

4.46

4.07

3.84

3.69

3.58

3.50

3.44

3.39

3.35

3.28

3.22

3.15

3.12

3.08

3.04

3.01

2.97

2.93

95.

124.

263.

863.

633.

483.

373.

293.

233.

183.

143.

073.

012.

942.

902.

862.

832.

792.

752.

7110

4.96

4.10

3.71

3.48

3.33

3.22

3.14

3.07

3.02

2.98

2.91

2.85

2.77

2.74

2.70

2.66

2.62

2.58

2.54

114.

843.

983.

593.

363.

203.

093.

012.

952.

902.

852.

792.

722.

652.

612.

572.

532.

492.

452.

4012

4.75

3.89

3.49

3.26

3.11

3.00

2.91

2.85

2.80

2.75

2.69

2.62

2.54

2.51

2.47

2.43

2.38

2.34

2.30

134.

673.

813.

413.

183.

032.

922.

832.

772.

712.

672.

602.

532.

462.

422.

382.

342.

302.

252.

2114

4.60

3.74

3.34

3.11

2.96

2.85

2.76

2.70

2.65

2.60

2.53

2.46

2.39

2.35

2.31

2.27

2.22

2.18

2.13

154.

543.

683.

293.

062.

902.

792.

712.

642.

592.

542.

482.

402.

332.

292.

252.

202.

162.

112.

0716

4.49

3.63

3.24

3.01

2.85

2.74

2.66

2.59

2.54

2.49

2.42

2.35

2.28

2.24

2.19

2.15

2.11

2.06

2.01

174.

453.

593.

202.

962.

812.

702.

612.

552.

492.

452.

382.

312.

232.

192.

152.

102.

062.

011.

9618

4.41

3.55

3.16

2.93

2.77

2.66

2.58

2.51

2.46

2.41

2.34

2.27

2.19

2.15

2.11

2.06

2.02

1.97

1.92

194.

383.

523.

132.

902.

742.

632.

542.

482.

422.

382.

312.

232.

162.

112.

072.

031.

981.

931.

8820

4.35

3.49

3.10

2.87

2.71

2.60

2.51

2.45

2.39

2.35

2.28

2.20

2.12

2.08

2.04

1.99

1.95

1.90

1.84

214.

323.

473.

072.

842.

682.

572.

492.

422.

372.

322.

252.

182.

102.

052.

011.

961.

921.

871.

8122

4.30

3.44

3.05

2.82

2.66

2.55

2.46

2.40

2.34

2.30

2.23

2.15

2.07

2.03

1.98

1.94

1.89

1.84

1.78

234.

283.

423.

032.

802.

642.

532.

442.

372.

322.

272.

202.

132.

052.

011.

961.

911.

861.

811.

7624

4.26

3.40

3.01

2.78

2.62

2.51

2.42

2.36

2.30

2.25

2.18

2.11

2.03

1.98

1.94

1.89

1.84

1.79

1.73

254.

243.

392.

992.

762.

602.

492.

402.

342.

282.

242.

162.

092.

011.

961.

921.

871.

821.

771.

7126

4.23

3.37

2.98

2.74

2.59

2.47

2.39

2.32

2.27

2.22

2.15

2.07

1.99

1.95

1.90

1.85

1.80

1.75

1.69

274.

213.

352.

962.

732.

572.

462.

372.

312.

252.

202.

132.

061.

971.

931.

881.

841.

791.

731.

6728

4.20

3.34

2.95

2.71

2.56

2.45

2.36

2.29

2.24

2.19

2.12

2.04

1.96

1.91

1.87

1.82

1.77

1.71

1.65

294.

183.

332.

932.

702.

552.

432.

352.

282.

222.

182.

102.

031.

941.

901.

851.

811.

751.

701.

6430

4.17

3.32

2.92

2.69

2.53

2.42

2.33

2.27

2.21

2.16

2.09

2.01

1.93

1.89

1.84

1.79

1.74

1.68

1.62

404.

083.

232.

842.

612.

452.

342.

252.

182.

122.

082.

001.

921.

841.

791.

741.

691.

641.

581.

5160

4.00

3.15

2.76

2.53

2.37

2.25

2.17

2.10

2.04

1.99

1.92

1.84

1.75

1.70

1.65

1.59

1.53

1.47

1.39

120

3.92

3.07

2.68

2.45

2.29

2.18

2.09

2.02

1.96

1.91

1.83

1.75

1.66

1.61

1.55

1.50

1.43

1.35

1.25

∞3.

843.

002.

602.

372.

212.

102.

011.

941.

881.

831.

751.

671.

571.

521.

461.

391.

321.

221.

00

24

Upp

er2.

5%po

ints

ν1

12

34

56

78

910

1215

2024

3040

6012

0∞

ν2 1

647.

879

9.5

864.

289

9.6

921.

893

7.1

948.

295

6.7

963.

396

8.6

976.

798

4.9

993.

199

7.2

1001

.410

05.6

1009

.810

14.0

1018

.32

38.5

139

.00

39.1

739

.25

39.3

039

.33

39.3

639

.37

39.3

939

.40

39.4

139

.43

39.4

539

.46

39.4

639

.47

39.4

839

.49

39.5

03

17.4

416

.04

15.4

415

.10

14.8

814

.73

14.6

214

.54

14.4

714

.42

14.3

414

.25

14.1

714

.12

14.0

814

.04

13.9

913

.95

13.9

04

12.2

210

.65

9.98

9.60

9.36

9.20

9.07

8.98

8.90

8.84

8.75

8.66

8.56

8.51

8.46

8.41

8.36

8.31

8.26

510

.01

8.43

7.76

7.39

7.15

6.98

6.85

6.76

6.68

6.62

6.52

6.43

6.33

6.28

6.23

6.18

6.12

6.07

6.02

68.

817.

266.

606.

235.

995.

825.

705.

605.

525.

465.

375.

275.

175.

125.

075.

014.

964.

904.

857

8.07

6.54

5.89

5.52

5.29

5.12

4.99

4.90

4.82

4.76

4.67

4.57

4.47

4.41

4.36

4.31

4.25

4.20

4.14

87.

576.

065.

425.

054.

824.

654.

534.

434.

364.

304.

204.

104.

003.

953.

893.

843.

783.

733.

679

7.21

5.71

5.08

4.72

4.48

4.32

4.20

4.10

4.03

3.96

3.87

3.77

3.67

3.61

3.56

3.51

3.45

3.39

3.33

106.

945.

464.

834.

474.

244.

073.

953.

853.

783.

723.

623.

523.

423.

373.

313.

263.

203.

143.

0811

6.72

5.26

4.63

4.28

4.04

3.88

3.76

3.66

3.59

3.53

3.43

3.33

3.23

3.17

3.12

3.06

3.00

2.94

2.88

126.

555.

104.

474.

123.

893.

733.

613.

513.

443.

373.

283.

183.

073.

022.

962.

912.

852.

792.

7213

6.41

4.97

4.35

4.00

3.77

3.60

3.48

3.39

3.31

3.25

3.15

3.05

2.95

2.89

2.84

2.78

2.72

2.66

2.60

146.

304.

864.

243.

893.

663.

503.

383.

293.

213.

153.

052.

952.

842.

792.

732.

672.

612.

552.

4915

6.20

4.77

4.15

3.80

3.58

3.41

3.29

3.20

3.12

3.06

2.96

2.86

2.76

2.70

2.64

2.59

2.52

2.46

2.40

166.

124.

694.

083.

733.

503.

343.

223.

123.

052.

992.

892.

792.

682.

632.

572.

512.

452.

382.

3217

6.04

4.62

4.01

3.66

3.44

3.28

3.16

3.06

2.98

2.92

2.82

2.72

2.62

2.56

2.50

2.44

2.38

2.32

2.25

185.

984.

563.

953.

613.

383.

223.

103.

012.

932.

872.

772.

672.

562.

502.

442.

382.

322.

262.

1919

5.92

4.51

3.90

3.56

3.33

3.17

3.05

2.96

2.88

2.82

2.72

2.62

2.51

2.45

2.39

2.33

2.27

2.20

2.13

205.

874.

463.

863.

513.

293.

133.

012.

912.

842.

772.

682.

572.

462.

412.

352.

292.

222.

162.

0921

5.83

4.42

3.82

3.48

3.25

3.09

2.97

2.87

2.80

2.73

2.64

2.53

2.42

2.37

2.31

2.25

2.18

2.11

2.04

225.

794.

383.

783.

443.

223.

052.

932.

842.

762.

702.

602.

502.

392.

332.

272.

212.

142.

082.

0023

5.75

4.35

3.75

3.41

3.18

3.02

2.90

2.81

2.73

2.67

2.57

2.47

2.36

2.30

2.24

2.18

2.11

2.04

1.97

245.

724.

323.

723.

383.

152.

992.

872.

782.

702.

642.

542.

442.

332.

272.

212.

152.

082.

011.

9425

5.69

4.29

3.69

3.35

3.13

2.97

2.85

2.75

2.68

2.61

2.51

2.41

2.30

2.24

2.18

2.12

2.05

1.98

1.91

265.

664.

273.

673.

333.

102.

942.

822.

732.

652.

592.

492.

392.

282.

222.

162.

092.

031.

951.

8827

5.63

4.24

3.65

3.31

3.08

2.92

2.80

2.71

2.63

2.57

2.47

2.36

2.25

2.19

2.13

2.07

2.00

1.93

1.85

285.

614.

223.

633.

293.

062.

902.

782.

692.

612.

552.

452.

342.

232.

172.

112.

051.

981.

911.

8329

5.59

4.20

3.61

3.27

3.04

2.88

2.76

2.67

2.59

2.53

2.43

2.32

2.21

2.15

2.09

2.03

1.96

1.89

1.81

305.

574.

183.

593.

253.

032.

872.

752.

652.

572.

512.

412.

312.

202.

142.

072.

011.

941.

871.

7940

5.42

4.05

3.46

3.13

2.90

2.74

2.62

2.53

2.45

2.39

2.29

2.18

2.07

2.01

1.94

1.88

1.80

1.72

1.64

605.

293.

933.

343.

012.

792.

632.

512.

412.

332.

272.

172.

061.

941.

881.

821.

741.

671.

581.

4812

05.

153.

803.

232.

892.

672.

522.

392.

302.

222.

162.

051.

941.

821.

761.

691.

611.

531.

431.

31∞

5.02

3.69

3.12

2.79

2.57

2.41

2.29

2.19

2.11

2.05

1.94

1.83

1.71

1.64

1.57

1.48

1.39

1.27

1.00

25

Upp

er1%

poin

tsν1

12

34

56

78

910

1215

2024

3040

6012

0∞

ν2 1

4052

4999

5403

5625

5764

5859

5928

5981

6022

6056

6106

6157

6209

6235

6261

6287

6313

6339

6366

298

.50

99.0

099

.17

99.2

599

.30

99.3

399

.36

99.3

799

.39

99.4

099

.42

99.4

399

.45

99.4

699

.47

99.4

799

.48

99.4

999

.50

334

.12

30.8

229

.46

28.7

128

.24

27.9

127

.67

27.4

927

.35

27.2

327

.05

26.8

726

.69

26.6

026

.50

26.4

126

.32

26.2

226

.13

421

.20

18.0

016

.69

15.9

815

.52

15.2

114

.98

14.8

014

.66

14.5

514

.37

14.2

014

.02

13.9

313

.84

13.7

513

.65

13.5

613

.46

516

.26

13.2

712

.06

11.3

910

.97

10.6

710

.46

10.2

910

.16

10.0

59.

899.

729.

559.

479.

389.

299.

209.

119.

026

13.7

510

.92

9.78

9.15

8.75

8.47

8.26

8.10

7.98

7.87

7.72

7.56

7.40

7.31

7.23

7.14

7.06

6.97

6.88

712

.25

9.55

8.45

7.85

7.46

7.19

6.99

6.84

6.72

6.62

6.47

6.31

6.16

6.07

5.99

5.91

5.82

5.74

5.65

811

.26

8.65

7.59

7.01

6.63

6.37

6.18

6.03

5.91

5.81

5.67

5.52

5.36

5.28

5.20

5.12

5.03

4.95

4.86

910

.56

8.02

6.99

6.42

6.06

5.80

5.61

5.47

5.35

5.26

5.11

4.96

4.81

4.73

4.65

4.57

4.48

4.40

4.31

1010

.04

7.56

6.55

5.99

5.64

5.39

5.20

5.06

4.94

4.85

4.71

4.56

4.41

4.33

4.25

4.17

4.08

4.00

3.91

119.

657.

216.

225.

675.

325.

074.

894.

744.

634.

544.

404.

254.

104.

023.

943.

863.

783.

693.

6012

9.33

6.93

5.95

5.41

5.06

4.82

4.64

4.50

4.39

4.30

4.16

4.01

3.86

3.78

3.70

3.62

3.54

3.45

3.36

139.

076.

705.

745.

214.

864.

624.

444.

304.

194.

103.

963.

823.

663.

593.

513.

433.

343.

253.

1714

8.86

6.51

5.56

5.04

4.69

4.46

4.28

4.14

4.03

3.94

3.80

3.66

3.51

3.43

3.35

3.27

3.18

3.09

3.00

158.

686.

365.

424.

894.

564.

324.

144.

003.

893.

803.

673.

523.

373.

293.

213.

133.

052.

962.

8716

8.53

6.23

5.29

4.77

4.44

4.20

4.03

3.89

3.78

3.69

3.55

3.41

3.26

3.18

3.10

3.02

2.93

2.84

2.75

178.

406.

115.

184.

674.

344.

103.

933.

793.

683.

593.

463.

313.

163.

083.

002.

922.

832.

752.

6518

8.29

6.01

5.09

4.58

4.25

4.01

3.84

3.71

3.60

3.51

3.37

3.23

3.08

3.00

2.92

2.84

2.75

2.66

2.57

198.

185.

935.

014.

504.

173.

943.

773.

633.

523.

433.

303.

153.

002.

922.

842.

762.

672.

582.

4920

8.10

5.85

4.94

4.43

4.10

3.87

3.70

3.56

3.46

3.37

3.23

3.09

2.94

2.86

2.78

2.69

2.61

2.52

2.42

218.

025.

784.

874.

374.

043.

813.

643.

513.

403.

313.

173.

032.

882.

802.

722.

642.

552.

462.

3622

7.95

5.72

4.82

4.31

3.99

3.76

3.59

3.45

3.35

3.26

3.12

2.98

2.83

2.75

2.67

2.58

2.50

2.40

2.31

237.

885.

664.

764.

263.

943.

713.

543.

413.

303.

213.

072.

932.

782.

702.

622.

542.

452.

352.

2624

7.82

5.61

4.72

4.22

3.90

3.67

3.50

3.36

3.26

3.17

3.03

2.89

2.74

2.66

2.58

2.49

2.40

2.31

2.21

257.

775.

574.

684.

183.

853.

633.

463.

323.

223.

132.

992.

852.

702.

622.

542.

452.

362.

272.

1726

7.72

5.53

4.64

4.14

3.82

3.59

3.42

3.29

3.18

3.09

2.96

2.81

2.66

2.58

2.50

2.42

2.33

2.23

2.13

277.

685.

494.

604.

113.

783.

563.

393.

263.

153.

062.

932.

782.

632.

552.

472.

382.

292.

202.

1028

7.64

5.45

4.57

4.07

3.75

3.53

3.36

3.23

3.12

3.03

2.90

2.75

2.60

2.52

2.44

2.35

2.26

2.17

2.06

297.

605.

424.

544.

043.

733.

503.

333.

203.

093.

002.

872.

732.

572.

492.

412.

332.

232.

142.

0330

7.56

5.39

4.51

4.02

3.70

3.47

3.30

3.17

3.07

2.98

2.84

2.70

2.55

2.47

2.39

2.30

2.21

2.11

2.01

407.

315.

184.

313.

833.

513.

293.

122.

992.

892.

802.

662.

522.

372.

292.

202.

112.

021.

921.

8060

7.08

4.98

4.13

3.65

3.34

3.12

2.95

2.82

2.72

2.63

2.50

2.35

2.20

2.12

2.03

1.94

1.84

1.73

1.60

120

6.85

4.79

3.95

3.48

3.17

2.96

2.79

2.66

2.56

2.47

2.34

2.19

2.03

1.95

1.86

1.76

1.66

1.53

1.38

∞6.

634.

613.

783.

323.

022.

802.

642.

512.

412.

322.

182.

041.

881.

791.

701.

591.

471.

321.

00

26

Upp

er0.

5%po

ints

ν1

12

34

56

78

910

1215

2024

3040

6012

0∞

ν2 1

1621

119

999

2161

522

500

2305

623

437

2371

523

925

2409

124

224

2442

624

630

2483

624

940

2504

425

148

2525

325

359

2546

42

198.

519

9.0

199.

219

9.2

199.

319

9.3

199.

419

9.4

199.

419

9.4

199.

419

9.4

199.

419

9.5

199.

519

9.5

199.

519

9.5

199.

53

55.5

549

.80

47.4

746

.19

45.3

944

.84

44.4

344

.13

43.8

843

.69

43.3

943

.08

42.7

842

.62

42.4

742

.31

42.1

541

.99

41.8

34

31.3

326

.28

24.2

623

.15

22.4

621

.97

21.6

221

.35

21.1

420

.97

20.7

020

.44

20.1

720

.03

19.8

919

.75

19.6

119

.47

19.3

25

22.7

818

.31

16.5

315

.56

14.9

414

.51

14.2

013

.96

13.7

713

.62

13.3

813

.15

12.9

012

.78

12.6

612

.53

12.4

012

.27

12.1

46

18.6

314

.54

12.9

212

.03

11.4

611

.07

10.7

910

.57

10.3

910

.25

10.0

39.

819.

599.

479.

369.

249.

129.

008.

887

16.2

412

.40

10.8

810

.05

9.52

9.16

8.89

8.68

8.51

8.38

8.18

7.97

7.75

7.64

7.53

7.42

7.31

7.19

7.08

814

.69

11.0

49.

608.

818.

307.

957.

697.

507.

347.

217.

016.

816.

616.

506.

406.

296.

186.

065.

959

13.6

110

.11

8.72

7.96

7.47

7.13

6.88

6.69

6.54

6.42

6.23

6.03

5.83

5.73

5.62

5.52

5.41

5.30

5.19

1012

.83

9.43

8.08

7.34

6.87

6.54

6.30

6.12

5.97

5.85

5.66

5.47

5.27

5.17

5.07

4.97

4.86

4.75

4.64

1112

.23

8.91

7.60

6.88

6.42

6.10

5.86

5.68

5.54

5.42

5.24

5.05

4.86

4.76

4.65

4.55

4.45

4.34

4.23

1211

.75

8.51

7.23

6.52

6.07

5.76

5.52

5.35

5.20

5.09

4.91

4.72

4.53

4.43

4.33

4.23

4.12

4.01

3.90

1311

.37

8.19

6.93

6.23

5.79

5.48

5.25

5.08

4.94

4.82

4.64

4.46

4.27

4.17

4.07

3.97

3.87

3.76

3.65

1411

.06

7.92

6.68

6.00

5.56

5.26

5.03

4.86

4.72

4.60

4.43

4.25

4.06

3.96

3.86

3.76

3.66

3.55

3.44

1510

.80

7.70

6.48

5.80

5.37

5.07

4.85

4.67

4.54

4.42

4.25

4.07

3.88

3.79

3.69

3.58

3.48

3.37

3.26

1610

.58

7.51

6.30

5.64

5.21

4.91

4.69

4.52

4.38

4.27

4.10

3.92

3.73

3.64

3.54

3.44

3.33

3.22

3.11

1710

.38

7.35

6.16

5.50

5.07

4.78

4.56

4.39

4.25

4.14

3.97

3.79

3.61

3.51

3.41

3.31

3.21

3.10

2.98

1810

.22

7.21

6.03

5.37

4.96

4.66

4.44

4.28

4.14

4.03

3.86

3.68

3.50

3.40

3.30

3.20

3.10

2.99

2.87

1910

.07

7.09

5.92

5.27

4.85

4.56

4.34

4.18

4.04

3.93

3.76

3.59

3.40

3.31

3.21

3.11

3.00

2.89

2.78

209.

946.

995.

825.

174.

764.

474.

264.

093.

963.

853.

683.

503.

323.

223.

123.

022.

922.

812.

6921

9.83

6.89

5.73

5.09

4.68

4.39

4.18

4.01

3.88

3.77

3.60

3.43

3.24

3.15

3.05

2.95

2.84

2.73

2.61

229.

736.

815.

655.

024.

614.

324.

113.

943.

813.

703.

543.

363.

183.

082.

982.

882.

772.

662.

5523

9.63

6.73

5.58

4.95

4.54

4.26

4.05

3.88

3.75

3.64

3.47

3.30

3.12

3.02

2.92

2.82

2.71

2.60

2.48

249.

556.

665.

524.

894.

494.

203.

993.

833.

693.

593.

423.

253.

062.

972.

872.

772.

662.

552.

4325

9.48

6.60

5.46

4.84

4.43

4.15

3.94

3.78

3.64

3.54

3.37

3.20

3.01

2.92

2.82

2.72

2.61

2.50

2.38

269.

416.

545.

414.

794.

384.

103.

893.

733.

603.

493.

333.

152.

972.

872.

772.

672.

562.

452.

3327

9.34

6.49

5.36

4.74

4.34

4.06

3.85

3.69

3.56

3.45

3.28

3.11

2.93

2.83

2.73

2.63

2.52

2.41

2.29

289.

286.

445.

324.

704.

304.

023.

813.

653.

523.

413.

253.

072.

892.

792.

692.

592.

482.

372.

2529

9.23

6.40

5.28

4.66

4.26

3.98

3.77

3.61

3.48

3.38

3.21

3.04

2.86

2.76

2.66

2.56

2.45

2.33

2.21

309.

186.

355.

244.

624.

233.

953.

743.

583.

453.

343.

183.

012.

822.

732.

632.

522.

422.

302.

1840

8.83

6.07

4.98

4.37

3.99

3.71

3.51

3.35

3.22

3.12

2.95

2.78

2.60

2.50

2.40

2.30

2.18

2.06

1.93

608.

495.

794.

734.

143.

763.

493.

293.

133.

012.

902.

742.

572.

392.

292.

192.

081.

961.

831.

6912

08.

185.

544.

503.

923.

553.

283.

092.

932.

812.

712.

542.

372.

192.

091.

981.

871.

751.

611.

43∞

7.88

5.30

4.28

3.72

3.35

3.09

2.90

2.74

2.62

2.52

2.36

2.19

2.00

1.90

1.79

1.67

1.53

1.36

1.00

27

21 Poisson Tables

Values of P (r) =µre−µ

r!µ

r 01

23

45

67

89

1011

1213

1415

16···

21

0.02

0.98

00.

020

0.04

0.96

10.

038

0.00

1

0.06

0.94

20.

057

0.00

2

0.08

0.92

30.

074

0.00

3

0.10

0.90

50.

090

0.00

5

0.15

0.86

10.

129

0.01

0

0.20

0.81

90.

164

0.01

60.

001

0.25

0.77

90.

195

0.02

40.

002

0.30

0.74

10.

222

0.03

30.

003

0.35

0.70

50.

247

0.04

30.

005

0.40

0.67

00.

268

0.05

40.

007

0.00

1

0.45

0.63

80.

287

0.06

50.

010

0.00

1

0.50

0.60

70.

303

0.07

60.

013

0.00

2

0.55

0.57

70.

317

0.08

70.

016

0.00

2

0.60

0.54

90.

329

0.09

90.

020

0.00

3

0.65

0.52

20.

339

0.11

00.

024

0.00

40.

001

0.70

0.49

70.

348

0.12

20.

028

0.00

50.

001

0.75

0.47

20.

354

0.13

30.

033

0.00

60.

001

0.80

0.44

90.

359

0.14

40.

038

0.00

80.

001

0.85

0.42

70.

363

0.15

40.

044

0.00

90.

002

0.90

0.40

70.

366

0.16

50.

049

0.01

10.

002

0.95

0.38

70.

367

0.17

50.

055

0.01

30.

002

1.00

0.36

80.

368

0.18

40.

061

0.01

50.

003

0.00

1

1.10

0.33

30.

366

0.20

10.

074

0.02

00.

004

0.00

1

1.20

0.30

10.

361

0.21

70.

087

0.02

60.

006

0.00

1

1.30

0.27

30.

354

0.23

00.

100

0.03

20.

008

0.00

2

1.40

0.24

70.

345

0.24

20.

113

0.03

90.

011

0.00

30.

001

1.50

0.22

30.

335

0.25

10.

126

0.04

70.

014

0.00

40.

001

1.60

0.20

20.

323

0.25

80.

138

0.05

50.

018

0.00

50.

001

28

µr 0

12

34

56

78

910

1112

1314

1516

1718

1920

21

1.70

0.18

30.

311

0.26

40.

150

0.06

40.

022

0.00

60.

001

1.80

0.16

50.

298

0.26

80.

161

0.07

20.

026

0.00

80.

002

1.90

0.15

00.

284

0.27

00.

171

0.08

10.

031

0.01

00.

003

0.00

1

2.00

0.13

50.

271

0.27

10.

180

0.09

00.

036

0.01

20.

003

0.00

1

2.10

0.12

20.

257

0.27

00.

189

0.09

90.

042

0.01

50.

004

0.00

1

2.20

0.11

10.

244

0.26

80.

197

0.10

80.

048

0.01

70.

005

0.00

2

2.30

0.10

00.

231

0.26

50.

203

0.11

70.

054

0.02

10.

007

0.00

2

2.40

0.09

10.

218

0.26

10.

209

0.12

50.

060

0.02

40.

008

0.00

20.

001

2.50

0.08

20.

205

0.25

70.

214

0.13

40.

067

0.02

80.

010

0.00

30.

001

2.60

0.07

40.

193

0.25

10.

218

0.14

10.

074

0.03

20.

012

0.00

40.

001

2.70

0.06

70.

181

0.24

50.

220

0.14

90.

080

0.03

60.

014

0.00

50.

001

2.80

0.06

10.

170

0.23

80.

222

0.15

60.

087

0.04

10.

016

0.00

60.

002

2.90

0.05

50.

160

0.23

10.

224

0.16

20.

094

0.04

50.

019

0.00

70.

002

0.00

1

3.00

0.05

00.

149

0.22

40.

224

0.16

80.

101

0.05

00.

022

0.00

80.

003

0.00

1

3.10

0.04

50.

140

0.21

60.

224

0.17

30.

107

0.05

60.

025

0.01

00.

003

0.00

1

3.20

0.04

10.

130

0.20

90.

223

0.17

80.

114

0.06

10.

028

0.01

10.

004

0.00

1

3.30

0.03

70.

122

0.20

10.

221

0.18

20.

120

0.06

60.

031

0.01

30.

005

0.00

2

3.40

0.03

30.

113

0.19

30.

219

0.18

60.

126

0.07

20.

035

0.01

50.

006

0.00

20.

001

3.50

0.03

00.

106

0.18

50.

216

0.18

90.

132

0.07

70.

039

0.01

70.

007

0.00

20.

001

3.60

0.02

70.

098

0.17

70.

212

0.19

10.

138

0.08

30.

042

0.01

90.

008

0.00

30.

001

3.70

0.02

50.

091

0.16

90.

209

0.19

30.

143

0.08

80.

047

0.02

20.

009

0.00

30.

001

3.80

0.02

20.

085

0.16

20.

205

0.19

40.

148

0.09

40.

051

0.02

40.

010

0.00

40.

001

3.90

0.02

00.

079

0.15

40.

200

0.19

50.

152

0.09

90.

055

0.02

70.

012

0.00

50.

002

0.00

1

29

µr 0

12

34

56

78

910

1112

1314

1516

1718

1920

21

4.00

0.01

80.

073

0.14

70.

195

0.19

50.

156

0.10

40.

060

0.03

00.

013

0.00

50.

002

0.00

1

4.10

0.01

70.

068

0.13

90.

190

0.19

50.

160

0.10

90.

064

0.03

30.

015

0.00

60.

002

0.00

1

4.20

0.01

50.

063

0.13

20.

185

0.19

40.

163

0.11

40.

069

0.03

60.

017

0.00

70.

003

0.00

1

4.30

0.01

40.

058

0.12

50.

180

0.19

30.

166

0.11

90.

073

0.03

90.

019

0.00

80.

003

0.00

1

4.40

0.01

20.

054

0.11

90.

174

0.19

20.

169

0.12

40.

078

0.04

30.

021

0.00

90.

004

0.00

1

4.50

0.01

10.

050

0.11

20.

169

0.19

00.

171

0.12

80.

082

0.04

60.

023

0.01

00.

004

0.00

20.

001

4.60

0.01

00.

046

0.10

60.

163

0.18

80.

173

0.13

20.

087

0.05

00.

026

0.01

20.

005

0.00

20.

001

4.70

0.00

90.

043

0.10

00.

157

0.18

50.

174

0.13

60.

091

0.05

40.

028

0.01

30.

006

0.00

20.

001

4.80

0.00

80.

040

0.09

50.

152

0.18

20.

175

0.14

00.

096

0.05

80.

031

0.01

50.

006

0.00

30.

001

4.90

0.00

70.

036

0.08

90.

146

0.17

90.

175

0.14

30.

100

0.06

10.

033

0.01

60.

007

0.00

30.

001

5.00

0.00

70.

034

0.08

40.

140

0.17

50.

175

0.14

60.

104

0.06

50.

036

0.01

80.

008

0.00

30.

001

5.10

0.00

60.

031

0.07

90.

135

0.17

20.

175

0.14

90.

109

0.06

90.

039

0.02

00.

009

0.00

40.

002

0.00

1

5.20

0.00

60.

029

0.07

50.

129

0.16

80.

175

0.15

10.

113

0.07

30.

042

0.02

20.

010

0.00

50.

002

0.00

1

5.30

0.00

50.

026

0.07

00.

124

0.16

40.

174

0.15

40.

116

0.07

70.

045

0.02

40.

012

0.00

50.

002

0.00

1

5.40

0.00

50.

024

0.06

60.

119

0.16

00.

173

0.15

60.

120

0.08

10.

049

0.02

60.

013

0.00

60.

002

0.00

1

5.50

0.00

40.

022

0.06

20.

113

0.15

60.

171

0.15

70.

123

0.08

50.

052

0.02

90.

014

0.00

70.

003

0.00

1

5.60

0.00

40.

021

0.05

80.

108

0.15

20.

170

0.15

80.

127

0.08

90.

055

0.03

10.

016

0.00

70.

003

0.00

1

5.70

0.00

30.

019

0.05

40.

103

0.14

70.

168

0.15

90.

130

0.09

20.

059

0.03

30.

017

0.00

80.

004

0.00

1

5.80

0.00

30.

018

0.05

10.

098

0.14

30.

166

0.16

00.

133

0.09

60.

062

0.03

60.

019

0.00

90.

004

0.00

20.

001

5.90

0.00

30.

016

0.04

80.

094

0.13

80.

163

0.16

00.

135

0.10

00.

065

0.03

90.

021

0.01

00.

005

0.00

20.

001

30

µr 0

12

34

56

78

910

1112

1314

1516

1718

1920

21

6.00

0.00

20.

015

0.04

50.

089

0.13

40.

161

0.16

10.

138

0.10

30.

069

0.04

10.

023

0.01

10.

005

0.00

20.

001

6.20

0.00

20.

013

0.03

90.

081

0.12

50.

155

0.16

00.

142

0.11

00.

076

0.04

70.

026

0.01

40.

007

0.00

30.

001

6.40

0.00

20.

011

0.03

40.

073

0.11

60.

149

0.15

90.

145

0.11

60.

082

0.05

30.

031

0.01

60.

008

0.00

40.

002

0.00

1

6.60

0.00

10.

009

0.03

00.

065

0.10

80.

142

0.15

60.

147

0.12

10.

089

0.05

90.

035

0.01

90.

010

0.00

50.

002

0.00

1

6.80

0.00

10.

008

0.02

60.

058

0.09

90.

135

0.15

30.

149

0.12

60.

095

0.06

50.

040

0.02

30.

012

0.00

60.

003

0.00

1

7.00

0.00

10.

006

0.02

20.

052

0.09

10.

128

0.14

90.

149

0.13

00.

101

0.07

10.

045

0.02

60.

014

0.00

70.

003

0.00

10.

001

7.20

0.00

10.

005

0.01

90.

046

0.08

40.

120

0.14

40.

149

0.13

40.

107

0.07

70.

050

0.03

00.

017

0.00

90.

004

0.00

20.

001

7.40

0.00

10.

005

0.01

70.

041

0.07

60.

113

0.13

90.

147

0.13

60.

112

0.08

30.

056

0.03

40.

020

0.01

00.

005

0.00

20.

001

7.60

0.00

10.

004

0.01

40.

037

0.07

00.

106

0.13

40.

145

0.13

80.

117

0.08

90.

061

0.03

90.

023

0.01

20.

006

0.00

30.

001

0.00

1

7.80

0.00

00.

003

0.01

20.

032

0.06

30.

099

0.12

80.

143

0.13

90.

121

0.09

40.

067

0.04

30.

026

0.01

50.

008

0.00

40.

002

0.00

1

8.00

0.00

00.

003

0.01

10.

029

0.05

70.

092

0.12

20.

140

0.14

00.

124

0.09

90.

072

0.04

80.

030

0.01

70.

009

0.00

50.

002

0.00

1

9.00

0.00

00.

001

0.00

50.

015

0.03

40.

061

0.09

10.

117

0.13

20.

132

0.11

90.

097

0.07

30.

050

0.03

20.

019

0.01

10.

006

0.00

30.

001

0.00

1

10.0

00.

000

0.00

00.

002

0.00

80.

019

0.03

80.

063

0.09

00.

113

0.12

50.

125

0.11

40.

095

0.07

30.

052

0.03

50.

022

0.01

30.

007

0.00

40.

002

0.00

1

31

22 Legendre Polynomials

Legendre polynomials under standard normalisation Pn(1) = 1 :

The Legendre polynomials are defined for x ∈ R by the two term recurrence relation,

P0(x) = 1,

P1(x) = x,

(n+ 1)Pn+1(x) = (2n+ 1)xPn(x)− nPn−1(x), n ∈ N.

The next two terms are

P2(x) =3

2x2 − 1

2, P3(x) =

5

2x3 − 3

2x.

Orthogonality properties on the interval [−1, 1] :

Consider the inner product 〈·, ·〉 defined by

〈f, g〉 =

∫ 1

−1

f(x)g(x) dx,

for continuous functions f, g : [−1, 1]→ R . The corresponding norm is given by

‖f‖ =√〈f, f〉.

The Legendre polynomials are orthogonal with respect to the inner product:

〈Pn, Pm〉 = 0, m 6= n.

Legendre polynomials with orthogonal normalisation:

Orthogonally normalised Legendre polynomials are defined by

φn =Pn(x)

‖Pn‖.

The first few terms are

φ0(x) =1√2, φ1(x) =

√3

2x, φ2(x) =

√5

2

(3

2x2 − 1

2

).

32

23 Orthogonal Polynomials (for equidistant abscissae)

n 3 4 5 6

fi f1 f2 f1 f2 f3 f1 f2 f3 f4 f1 f2 f3 f4 f5

−5 +5 −5 +1 −1−2 +2 −1 +1

−3 +1 −1 −3 −1 +7 −3 +5−1 +1 −1 −1 +2 −4

−1 −1 +3 −1 −4 +4 +2 −10

0 −2 0 −2 0 +6+1 −1 −3 +1 −4 −4 +2 +10

+1 +1 +1 −1 −2 −4+3 +1 +1 +3 −1 −7 −3 −5

+2 +2 +1 +1+5 +5 +5 +1 +1∑

f 2i 2 6 20 4 20 10 14 10 70 70 84 180 28 252

λi 1 3 2 1 103

1 1 56

3512

2 32

53

712

2110

n 7 8 9

fi f1 f2 f3 f4 f5 f1 f2 f3 f4 f5 f1 f2 f3 f4 f5

0 −4 0 +6 0 0 −20 0 +18 0+1 −5 −3 +9 +15

+1 −3 −1 +1 +5 +1 −17 −9 +9 +9+3 −3 −7 −3 +17

+2 0 −1 −7 −4 +2 −8 −13 −11 +4+5 +1 −5 −13 −23

+3 +5 +1 +3 +1 +3 +7 −7 −21 −11+7 +7 +7 +7 +7

+4 +28 +14 +14 +4∑f 2i 28 84 6 154 84 168 168 264 616 2184 60 2772 990 2002 464

λi 1 1 16

712

720

2 1 23

712

710

1 3 56

712

320

f1(x) = λ1(x)

f2(x) = λ2

{x2 − 1

12(n2 − 1)

}

f3(x) = λ3

{x3 − 1

20(3n2 − 7)x

}

f4(x) = λ4

{x4 − 1

14(3n2 − 13)x2 +

3

560(n2 − 1)(n2 − 9)

}

f5(x) = λ5

{x5 − 5

18(n2 − 7)x3 +

1

1008(15n4 − 230n2 + 407)x

}

33

24 Random Numbers

The below table presents a typical series of random numbers for the convenience of classexercises. For practical work, reference should be made to a more extensive series such asthat in the Fisher and Yates statistical tables.

99050 30876 80821 14955 11495

08090 84688 36332 86858 73763

67619 00352 32735 59654 97851

63779 66008 02516 93874 67930

03259 72119 04769 95593 02754

92914 02066 97320 00328 51685

80001 70542 01530 63033 64384

37815 09824 86504 14817 74434

15897 74758 12779 69608 76893

06193 94893 24598 02714 69670

40134 12803 33942 46600 05681

88480 27598 48458 65639 08810

49989 94369 80429 97152 67613

62089 52111 92190 85413 95362

01675 12741 94334 86069 71353

04259 19768 47711 63262 06316

63859 63087 91886 43467 55595

17709 21642 56384 85699 24310

11727 83872 22553 17012 02949

02838 03160 92864 23985 63585

34

25 Wilcoxon Matched-Pairs Test

Critical values of T at Various Levels of Probability

Level of significance for two-tailed testN 0.10 0.05 0.02 0.01

5 0 – – –6 2 0 – –7 3 2 0 –8 5 3 1 09 8 5 3 1

10 10 8 5 311 13 10 7 512 17 13 9 713 21 17 12 914 25 21 15 1215 30 25 19 1516 35 29 23 1917 41 34 27 2318 47 40 32 2719 53 46 37 3220 60 52 43 3721 67 58 49 4222 75 65 55 4823 83 73 62 5424 91 81 69 6125 100 89 76 6826 110 98 84 7527 119 107 92 8328 130 116 101 9129 140 126 110 10030 151 137 120 10931 163 147 130 11832 175 159 140 12833 187 170 151 13834 200 182 162 14835 213 195 173 159

35

26 Mann-Whitney Test

1. Critical values of U for a One-tailed Test at α = 0.05 or a Two-tailed Test at α = 0.10

n1

n2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20123 04 0 15 0 1 2 46 0 2 3 5 77 0 2 4 6 8 118 1 3 5 8 10 13 159 1 3 6 9 12 15 18 2110 1 4 7 11 14 17 20 24 2711 1 5 8 12 16 19 23 27 31 3412 2 5 9 13 17 21 26 30 34 38 4213 2 6 10 15 19 24 28 33 37 42 47 5114 2 7 11 16 21 26 31 36 41 46 51 56 6115 3 7 12 18 23 28 33 39 44 50 55 61 66 7216 3 8 14 19 25 30 36 42 48 54 60 65 71 77 8317 3 9 15 20 26 33 39 45 51 57 64 70 77 83 89 9618 4 9 16 22 28 35 41 48 55 61 68 75 82 88 95 102 10919 0 4 10 17 23 30 37 44 51 58 65 72 80 87 94 101 109 116 12320 0 4 11 18 25 32 39 47 54 62 69 77 84 92 100 107 115 123 130 138

2. Critical values of U for a One-tailed Test at α = 0.025 or a Two-tailed Test at α = 0.05

n1

n2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 201234 05 0 1 26 1 2 3 57 1 3 5 6 88 0 2 4 6 8 10 139 0 2 4 7 10 12 15 1710 0 3 5 8 11 14 17 20 2311 0 3 6 9 13 16 19 23 26 3012 1 4 7 11 14 18 22 26 29 33 3713 1 4 8 12 16 20 24 28 33 37 41 4514 1 5 9 13 17 22 26 31 36 40 45 50 5515 1 5 10 14 19 24 29 34 39 44 49 54 59 6416 1 6 11 15 21 26 31 37 42 47 53 59 64 70 7517 2 6 11 17 22 28 34 39 45 51 57 63 67 75 81 8718 2 7 12 18 24 30 36 42 48 55 61 67 74 80 86 93 9919 2 7 13 19 25 32 38 45 52 58 65 72 78 85 92 99 106 11320 2 8 13 20 27 34 41 48 55 62 69 76 83 90 98 105 112 119 127

36

Mann-Whitney Test (continued)

3. Critical values of U for a One-tailed Test at α = 0.01 or a Two-tailed Test at α = 0.02

n1

n2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2012345 0 16 1 2 37 0 1 3 4 68 0 2 4 6 8 109 1 3 5 7 9 11 1410 1 3 6 8 11 13 16 1911 1 4 7 9 12 15 18 22 2512 2 5 8 11 14 17 21 24 28 3113 0 2 5 9 12 16 20 23 27 31 35 3914 0 2 6 10 13 17 22 26 30 34 38 43 4715 0 3 7 11 15 19 24 28 33 37 42 47 51 5616 0 3 7 12 16 21 26 31 36 41 46 51 56 61 6617 0 4 8 13 18 23 28 33 38 44 49 55 60 66 71 7718 0 4 9 14 19 24 30 36 41 47 53 59 65 70 76 82 8819 1 4 9 15 20 26 32 38 44 50 56 63 69 75 82 88 94 10120 1 5 10 16 22 28 34 40 47 53 60 67 73 80 87 93 100 107 114

37

27 Rank Correlation Coefficients (Spearman’s)

Critical Values of r

Level of significance for two-tailed testn 0.10 0.05 0.02 0.01

5 0.900 1.000 1.000 –6 0.829 0.886 0.943 1.0007 0.714 0.786 0.893 0.9298 0.643 0.738 0.833 0.8819 0.600 0.683 0.783 0.833

10 0.564 0.648 0.746 0.794

12 0.506 0.591 0.712 0.77714 0.456 0.544 0.645 0.71516 0.425 0.506 0.601 0.66518 0.399 0.475 0.564 0.62520 0.377 0.450 0.534 0.59122 0.359 0.428 0.508 0.56224 0.343 0.409 0.485 0.53726 0.329 0.392 0.465 0.51528 0.317 0.377 0.448 0.49630 0.306 0.364 0.432 0.478

28 Correlation CoefficientsCritical Values of r

Level of significance for two-tailed testn 0.10 0.05 0.02 0.01

4 0.900 0.950 0.980 0.9905 0.805 0.878 0.934 0.9596 0.729 0.811 0.882 0.9177 0.669 0.754 0.833 0.8748 0.621 0.707 0.789 0.8349 0.582 0.666 0.750 0.798

10 0.549 0.632 0.716 0.765

12 0.497 0.576 0.658 0.70814 0.457 0.532 0.612 0.66116 0.426 0.497 0.574 0.62318 0.400 0.468 0.543 0.59020 0.378 0.444 0.516 0.561

25 0.337 0.397 0.463 0.50730 0.308 0.361 0.423 0.46435 0.283 0.335 0.392 0.43040 0.264 0.312 0.367 0.40350 0.235 0.279 0.328 0.361

38

29 Constants for Use in Constructing Quality Control Charts

A0.025 = 1.96an/√

n . A0.001 = 3.1an/√

n .

Control limits at x ± Aαw where w is the average sample range when system is undercontrol.

Prob(range < Dαw

)= α

No. in Chart for means Chart for ranges

Sample Factors for control limits σ = anω Factors for control limits

n A0.025 A0.001 an D0.95 D0.995 D0.999 F0.95

2 1.23 1.94 0.8862 2.45 3.52 4.12 0.08

3 0.67 1.05 0.5908 1.96 2.58 2.98 0.25

4 0.48 0.75 0.4857 1.76 2.26 2.57 0.37

5 0.37 0.59 0.4299 1.66 2.08 2.34 0.44

6 0.32 0.50 0.3946 1.59 1.97 2.21 0.49

7 0.27 0.43 0.3698 1.54 1.90 2.11 0.53

8 0.24 0.38 0.3512 1.51 1.84 2.04 0.56

9 0.22 0.35 0.3367 1.48 1.79 1.99 0.59

10 0.20 0.32 0.3249 1.45 1.75 1.93 0.60

39

30 Some Common Families of Distributions

Discrete Distributions

Distribution Point probability Mean Variance Probabilitygeneratingfunction

Binomial (n, p)(nr

)pr(1− p)n−r np np(1− p) (1− p+ pz)n

r = 0, 1, 2, · · ·n

Poisson (λ) e−λλr/r! λ λ eλ(z−1)

r = 0, 1, 2, · · ·

Negative-(k+r−1

r

)pk(1− p)r k(1−p)

pk(1−p)p2

(p

1−z+pz)k

Binomial (k, p) r = 0, 1, 2, · · ·

Hyper-(N1

r

)(N2

n−r)/(

N1+N2

n

)nN1

N1+N2

nN1N2(N1+N2−n)(N1+N2)2(N1+N2−1)

geometric (N1, N2, n) r = 0, 1, 2, · · · ,min(n,N1);

N1 < N2.

Continuous Distributions

Distribution Density Function Mean Variance Momentgeneratingfunction

Uniform(a, b) 1b−a , (a < x < b)

12(a+ b) 1

12(a− b)2 ebt−eat(b−a)t

Beta(r, s)Γ(r+s)xr−1(1−x)s−1

Γ(r)Γ(s)rr+2

rs(r+s)2(r+s+1) —

Gamma (s, α) αsxs−1e−αx

Γ(s) , (x > 0)sα

sα2

(αα−t

)s

Exponential(α) is the same as Gamma (1, α)

Normal (µ, σ2) 1σ√

2πe−

12(x−µσ )

2

µ σ2 eµt+12σ2t2

p -variate normal distribution ( µ,Σ )

Density function(2π)−

12p|Σ|− 1

2 e−12{(x−µ)TΣ−1(x−µ)} if Σ−1 exists

Mean µ , Variance Σ , Moment Generating Function e(tTµ+ 12tTΣt) .

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