Mathematical Formula Handbook[1]

7/30/2019 Mathematical Formula Handbook[1]

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Mathematical Formula Handbook


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Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Bibliography; Physical Constants

1. Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Arithmetic and Geometric progressions; Convergence of series: the ratio test;Convergence of series: the comparison test; Binomial expansion; Taylor and Maclaurin Series;

Power series with real variables; Integer series; Plane wave expansion2. Vector Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Scalar product; Equation of a line; Equation of a plane; Vector product; Scalar triple product;Vector triple product; Non-orthogonal basis; Summation convention

3. Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Unit matrices; Products; Transpose matrices; Inverse matrices; Determinants; 22 matrices;Product rules; Orthogonal matrices; Solving sets of linear simultaneous equations; Hermitian matrices;Eigenvalues and eigenvectors; Commutators; Hermitian algebra; Pauli spin matrices

4. Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Notation; Identities; Grad, Div, Curl and the Laplacian; Transformation of integrals

5. Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Complex numbers; De Moivres theorem; Power series for complex variables.

6. Trigonometric Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Relations between sides and angles of any plane triangle;Relations between sides and angles of any spherical triangle

7. Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Relations of the functions; Inverse functions

8. Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

9. Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

10. Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13Standard forms; Standard substitutions; Integration by parts; Differentiation of an integral;Dirac -function; Reduction formulae

11. Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Diffusion (conduction) equation; Wave equation; Legendres equation; Bessels equation;Laplaces equation; Spherical harmonics

12. Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

13. Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Taylor series for two variables; Stationary points; Changing variables: the chain rule;Changing variables in surface and volume integrals Jacobians

14. Fourier Series and Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Fourier series; Fourier series for other ranges; Fourier series for odd and even functions;Complex form of Fourier series; Discrete Fourier series; Fourier transforms; Convolution theorem;Parsevals theorem; Fourier transforms in two dimensions; Fourier transforms in three dimensions

15. Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

16. Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Finding the zeros of equations; Numerical integration of differential equations;Central difference notation; Approximating to derivatives; Interpolation: Everetts formula;Numerical evaluation of definite integrals

17. Treatment of Random Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Range method; Combination of errors

18. Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Mean and Variance; Probability distributions; Weighted sums of random variables;Statistics of a data sample x1, . . . , xn; Regression (least squares fitting)


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Introduction

This Mathematical Formaulae handbook has been prepared in response to a request from the Physics ConsultativeCommittee, with the hope that it will be useful to those studying physics. It is to some extent modelled on a similardocument issued by the Department of Engineering, but obviously reflects the particular interests of physicists.There was discussion as to whether it should also include physical formulae such as Maxwells equations, etc., buta decision was taken against this, partly on the grounds that the book would become unduly bulky, but mainlybecause, in its present form, clean copies can be made available to candidates in exams.

There has been wide consultation among the staff about the contents of this document, but inevitably some userswill seek in vain for a formula they feel strongly should be included. Please send suggestions for amendments tothe Secretary of the Teaching Committee, and they will be considered for incorporation in the next edition. TheSecretary will also be grateful to be informed of any (equally inevitable) errors which are found.

This book was compiled by Dr John Shakeshaft and typeset originally by Fergus Gallagher, and currently byDr Dave Green, using the TEX typesetting package.

Version 1.5 December 2005.

Bibliography

Abramowitz, M. & Stegun, I.A., Handbook of Mathematical Functions, Dover, 1965.

Gradshteyn, I.S. & Ryzhik, I.M., Table of Integrals, Series and Products, Academic Press, 1980.

Jahnke, E. & Emde, F., Tables of Functions, Dover, 1986.

Nordling, C. & Osterman, J., Physics Handbook, Chartwell-Bratt, Bromley, 1980.

Speigel, M.R., Mathematical Handbook of Formulas and Tables.(Schaums Outline Series, McGraw-Hill, 1968).

Physical Constants

Based on the Review of Particle Properties, Barnett et al., 1996, Physics Review D, 54, p1, and The FundamentalPhysical Constants, Cohen & Taylor, 1997, Physics Today, BG7. (The figures in parentheses give the 1-standard-deviation uncertainties in the last digits.)

speed of light in a vacuum c 2997 924 58 108 m s1 (by definition)permeability of a vacuum 0 4 10

7 H m1 (by definition)permittivity of a vacuum 0 1/0c

2 = 8854 187 817 . . . 1012 F m1

elementary charge e 1602 177 33(49) 1019 CPlanck constant h 6626 075 5(40) 1034 J sh/2 h 1054 572 66(63) 1034 J sAvogadro constant NA 6022 136 7(36) 10

23 mol1

unified atomic mass constant mu 1660 540 2(10) 1027 kgmass of electron me 9109 389 7(54) 10

31 kgmass of proton mp 1672 623 1(10) 10

27 kgBohr magneton eh/4me B 9274 015 4(31) 10

24 J T1

molar gas constant R 8314 510(70) J K1 mol1

Boltzmann constant kB 1380 658(12) 1023 J K1

StefanBoltzmann constant 5670 51(19) 108 W m2 K4

gravitational constant G 6672 59(85) 1011 N m2 kg2Other data

acceleration of free fall g 9806 65 m s2 (standard value at sea level)

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1. Series

Arithmetic and Geometric progressions

A.P. Sn = a + (a + d) + (a + 2d) + + [a + (n 1)d] = n2

[2a + (n 1)d]

G.P. Sn = a + ar + ar2 + + arn1 = a 1 r

n

1

r,

S =

a

1

rfor |r| < 1

(These results also hold for complex series.)

Convergence of series: the ratio test

Sn = u1 + u2 + u3 + + un converges as n if limn

un+1un < 1

Convergence of series: the comparison test

If each term in a series of positive terms is less than the corresponding term in a series known to be convergent,then the given series is also convergent.

Binomial expansion

(1 + x)n = 1 + nx +n(n 1)

2!x2 +

n(n 1)(n 2)3!

x3 +

If n is a positive integer the series terminates and is valid for all x: the term in xr is nCrxr or

nr

where nCr

n!

r!(n r)! is the number of different ways in which an unordered sample of r objects can be selected from a set ofn objects without replacement. When n is not a positive integer, the series does not terminate: the infinite series isconvergent for |x| < 1.

Taylor and Maclaurin SeriesIfy(x) is well-behaved in the vicinity ofx = a then it has a Taylor series,

y(x) = y(a + u) = y(a) + udy

dx+

u2

2!

d2y

dx2+

u3

3!

d3y

dx3+

where u = x a and the differential coefficients are evaluated at x = a. A Maclaurin series is a Taylor series witha = 0,

y(x) = y(0) + xdy

dx+

x2

2!

d2y

dx2+

x3

3!

d3y

dx3+

Power series with real variables

ex = 1 + x + x22!

+ + xnn!

+ valid for all x

ln(1 + x) = x x2

2+

x3

3+ + (1)n+1 x

n

n+ valid for 1 < x 1

cos x =eix + eix

2= 1 x

2

2!+

x4

4! x

6

6!+ valid for all values of x

sin x =eix eix

2i= x x

3

3!+

x5

5!+ valid for all values of x

tan x = x +1

3x3 +

2

15x5 + valid for

2< x 1, Z

0 xp

ex

dx = pZ0 x

p1

ex

dx = p!. ( 1/2)! = , ( 1/2)! = /2, etc.For any p, q > 1,

Z10

xp(1 x)q dx = p!q!(p + q + 1)!

.

Trigonometrical

Ifm, n are integers,Z

/2

0sinm cosnd=

m 1m + n

Z/2

0sinm2 cosnd=

n 1m + n

Z/2

0sinm cosn2d

and can therefore be reduced eventually to one of the following integrals

Z/2

0sin cosd= 1

2,

Z/2

0sind= 1,

Z/2

0cosd= 1,

Z/2

0d=

2.

Other

If In =Z

0xn exp(x2) dx then In = (n 1)

2In2, I0 =

1

2

, I1 =

1

2.

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11. Differential Equations

Diffusion (conduction) equation

t= 2

Wave equation

2 = 1c2

2

t2

Legendres equation

(1 x2) d2y

dx2 2x dy

dx+ l(l + 1)y = 0,

solutions of which are Legendre polynomials Pl(x), where Pl(x) =1

2ll!

d

dx

l

x2 1

l, Rodrigues formula so

P0(x) = 1, P1(x) = x, P2(x) = 12 (3x2 1) etc.

Recursion relation

Pl(x) =1

l[(2l 1)xPl1(x) (l 1)Pl2(x)]

Orthogonality

Z11

Pl(x)Pl(x) dx =2

2l + 1ll

Bessels equation

x2d2y

dx2+ x

dy

dx+ (x2 m2)y = 0,

solutions of which are Bessel functions Jm(x) of order m.

Series form of Bessel functions of the first kind

Jm(x) =

k=0

(1)k(x/2)m+2kk!(m + k)!

(integer m).

The same general form holds for non-integer m > 0.

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Laplaces equation

2u = 0If expressed in two-dimensional polar coordinates (see section 4), a solution is

u(,) =An + Bn

C exp(in) + D exp(in)

where A, B, C, D are constants and n is a real integer.

If expressed in three-dimensional polar coordinates (see section 4) a solution is

u(r,,) = Arl + Br(l+1)Pml C sin m + D cos mwhere l and m are integers with l |m| 0; A, B, C, D are constants;

Pml (cos) = sin|m|

d

d(cos)

|m|Pl(cos)

is the associated Legendre polynomial.

P0l (1) = 1.

If expressed in cylindrical polar coordinates (see section 4), a solution is

u(,, z) = Jm(n)A cos m + B sin mC exp(nz) + D exp(nz)where m and n are integers; A, B, C, D are constants.Spherical harmonics

The normalized solutions Yml (,) of the equation1

sin

sin

+

1

sin2

2

2

Yml + l(l + 1)Y

ml = 0

are called spherical harmonics, and have values given by

Yml (,) =

2l + 1

4

(l |m|)!(l + |m|)! P

ml (cos) e

im

(1)m for m 01 for m < 0

i.e., Y00 =1

4, Y01 =

34

cos, Y11 = 38 sin ei, etc.Orthogonality

Z4

Yml Ym

l d = ll mm

12. Calculus of Variations

The condition for I =Zba

F(y, y, x) dx to have a stationary value isF

y=

d

dx

F

y

, where y =

dy

dx. This is the

EulerLagrange equation.

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Changing variables in surface and volume integrals Jacobians

If an area A in the x, y plane maps into an area A in the u, v plane then

ZA

f(x, y) dx dy =ZA

f(u, v)J du dv where J =

x

u

x

vy

u

y

v

The Jacobian J is also written as (x, y)(u, v) . The corresponding formula for volume integrals is

ZV

f(x, y, z) dx dy dz =ZV

f(u, v, w)J du dv dw where now J =

x

u

x

v

x

wy

u

y

v

y

wz

u

z

v

z

w

14. Fourier Series and Transforms

Fourier series

Ify(x) is a function defined in the range x then

y(x) c0 +M

m=1

cm cos mx +M

m=1

sm sin mx

where the coefficients are

c0 =1

2Z

y(x) dx

cm =1

Z

y(x) cos mx dx (m = 1 , . . . , M)

sm =1

Z

y(x) sin mx dx (m = 1 , . . . , M)

with convergence to y(x) as M, M for all points where y(x) is continuous.

Fourier series for other ranges

Variable t, range 0 t T, (i.e., a periodic function of time with period T, frequency = 2/T).y(t) c0 + cm cos mt + sm sin mt

where

c0 =

2

ZT0

y(t) dt, cm =

ZT0

y(t) cos mt dt, sm =

ZT0

y(t) sin mt dt.

Variable x, range 0 x L,y(x) c0 + cm cos

2mx

L+ sm sin

2mx

L

where

c0 =1

L

ZL0

y(x) dx, cm =2

L

ZL0

y(x) cos2mx

Ldx, sm =

2

L

ZL0

y(x) sin2mx

Ldx.

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Fourier series for odd and even functions

If y(x) is an odd (anti-symmetric) function [i.e., y(x) = y(x)] defined in the range x , then onlysines are required in the Fourier series and sm =

2

Z

0y(x) sin mx dx. If, in addition, y(x) is symmetric about

x = /2, then the coefficients sm are given by sm = 0 (for m even), sm =4

Z/2

0y(x) sin mx dx (for m odd). If

y(x) is an even (symmetric) function [i.e., y(x) = y(x)] defined in the range x , then only constantand cosine terms are required in the Fourier series and c0 =

1

Z

0

y(x) dx, cm =2

Z

0

y(x) cos mx dx. If, in

addition, y(x) is anti-symmetric about x =

2, then c0 = 0 and the coefficients cm are given by cm = 0 (for m even),

cm =4

Z/2

0y(x) cos mx dx (for m odd).

[These results also apply to Fourier series with more general ranges provided appropriate changes are made to thelimits of integration.]

Complex form of Fourier series

Ify(x) is a function defined in the range x then

y(x) M

M Cm eimx

, Cm =

1

2Z

y(x) eimx

dx

with m taking all integer values in the range M. This approximation converges to y(x) as M under the sameconditions as the real form.

For other ranges the formulae are:

Variable t, range 0 t T, frequency = 2/T,

y(t) =

Cm eimt, Cm =

2

ZT0

y(t) eimt dt.

Variable x, range 0 x L,

y(x) =

Cm ei2mx/L, Cm =

1

L

ZL0

y(x) ei2mx/L dx.

Discrete Fourier series

If y(x) is a function defined in the range x which is sampled in the 2 N equally spaced points xn =nx/N [n = (N 1) . . . N], then

y(xn) = c0 + c1 cos xn + c2 cos2xn + + cN1 cos(N 1)xn + cNcos Nxn+ s1 sin xn + s2 sin 2xn + + sN1 sin(N 1)xn + sNsin Nxn

where the coefficients are

c0 =1

2Ny(xn)

cm =

1

Ny(xn) cos mxn (m = 1 , . . . , N 1)cN =

1

2Ny(xn) cos Nxn

sm =1

Ny(xn) sin mxn (m = 1 , . . . , N 1)

sN =1

2Ny(xn) sin Nxn

each summation being over the 2Nsampling points xn.

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Fourier transforms

Ify(x) is a function defined in the range x then the Fourier transformy() is defined by the equationsy(t) =

1

2

Z

y() eit d, y() = Z

y(t) eit dt.

If is replaced by 2f, where f is the frequency, this relationship becomes

y(t) =Z

y(f) ei2f t df,

y(f) =

Z

y(t) ei2f t dt.

Ify(t) is symmetric about t = 0 then

y(t) =1

Z

0y() cost d, y() = 2 Z

0y(t) cost dt.

Ify(t) is anti-symmetric about t = 0 then

y(t) =1

Z

0y() sint d, y() = 2 Z

0y(t) sint dt.

Specific cases

y(t) = a, |t| = 0, |t| >

(Top Hat), y() = 2a sin

2asinc()

where sinc(x) =sin(x)

x

y(t) = a(1 |t|/), |t| = 0, |t| >

(Saw-tooth), y() = 2a

2(1 cos) = asinc2

2

y(t) = exp(t2/t20) (Gaussian), y() = t0exp 2t20/4

y(t) = f(t) ei0t (modulated function), y() = f( 0)y(t) =

m=

(t m) (sampling function)

y() =

n=

( 2n/)

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24/28

Convolution theorem

Ifz(t) =Z

x()y(t ) d=

Z

x(t )y() d x(t) y(t) then z() =x()y().

Conversely, xy =x y.Parsevals theorem

Z

y(t) y(t) dt =

1

2Z

y()y() d (ify is normalised as on page 21)Fourier transforms in two dimensions

V(k) = Z V(r) eikr d2r=

Z

02rV(r)J0(kr) dr if azimuthally symmetric

Fourier transforms in three dimensionsExamples

V(r)

V(k)

1

4r

1

k2er

4r

1

k2 + 2

V(r) ikV(k)2V(r) k2V(k)

V(k) =

ZV(r) eikr d3r

= 4k

Z0

V(r) r sin kr dr if spherically symmetric

V(r) =1

(2)3

Z V(k) eikr d3k

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25/28

15. Laplace Transforms

Ify(t) is a function defined for t 0, the Laplace transform y(s) is defined by the equationy(s) = L{y(t)} =

Z

0esty(t) dt

Function y(t) (t > 0) Transform y(s)

(t) 1 Delta function

(t)1

sUnit step function

tnn!

sn+1

t1/2

1

2

s3

t1/2

s

eat 1(s + a)

sint

(s2 +2

costs

(s2 +2)

sinht

(s2 2)cosht

s

(s2 2)

eat

y(t) y(s + a)y(t ) (t ) es y(s)

ty(t) dyds

dy

dtsy(s) y(0)

dny

dtnsny(s) sn1y(0) sn2

dy

dt

0

dn1ydtn1

0

Zt0

y() dy(s)

sZt0

x() y(t ) dZt0

x(t ) y() d

x(s) y(s) Convolution theorem[Note that if y(t) = 0 for t < 0 then the Fourier transform of y(t) isy() = y(i).]

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16. Numerical Analysis

Finding the zeros of equations

If the equation is y = f(x) and xn is an approximation to the root then either

xn+1 = xn f(xn)f(xn)

. (Newton)

or, xn+1 = xn xn xn1f(xn) f(xn1) f(xn) (Linear interpolation)

are, in general, better approximations.

Numerical integration of differential equations

Ifdy

dx= f(x, y) then

yn+1 = yn + h f(xn, yn) where h = xn+1 xn (Euler method)Putting yn+1 = yn + h f(xn, yn) (improved Euler method)

then yn+1 = yn + h[f(xn, yn) + f(xn+1, yn+1)]2

Central difference notation

Ify(x) is tabulated at equal intervals ofx, where h is the interval, then yn+1/2 = yn+1 yn and2yn = yn+1/2 yn1/2

Approximating to derivativesdy

dx

n

yn+1 ynh

yn yn1h

yn+ 1/2 + yn 1/22h

where h = xn+1 xnd2y

dx2

n

yn+1 2yn + yn1h2

=2ynh2

Interpolation: Everetts formula

y(x) = y(x0 +h) y0 + y1 + 13!(

2 1)2y0 + 13!(2 1)2y1 +

where is the fraction of the interval h (= xn+1 xn) between the sampling points and = 1 . The first twoterms represent linear interpolation.

Numerical evaluation of definite integrals

Trapezoidal rule

The interval of integration is divided into n equal sub-intervals, each of width h; thenZba

f(x) dx h

c1

2f(a) + f(x1) + + f(xj) + + 1

2f(b)

where h = (b a)/n and xj = a + jh.

Simpsons rule

The interval of integration is divided into an even number (say 2n) of equal sub-intervals, each of width h =(b a)/2n; thenZb

af(x) dx h

3

f(a) + 4f(x1) + 2f(x2) + 4f(x3) + + 2f(x2n2) + 4f(x2n1) + f(b)

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Gausss integration formulae

These have the general formZ11

y(x) dx n

1

ciy(xi)

For n = 2 : xi = 05773; ci = 1, 1 (exact for any cubic).For n = 3 : xi = 07746, 00, 07746; ci = 0555,0888, 0555 (exact for any quintic).

17. Treatment of Random Errors

Sample mean x =1

n(x1 + x2 + xn)

Residual: d = x xStandard deviation of sample: s =

1n

(d21 + d22 + d2n)1/2

Standard deviation of distribution:

1

n 1(d21 + d

22 +

d2n)

1/2

Standard deviation of mean: m =

n=

1n(n 1)

(d21 + d22 + d2n)1/2

=1

n(n 1)

x

2i

1

n

xi

21/2

Result ofn measurements is quoted as x m.

Range method

A quick but crude method of estimating is to find the range r of a set of n readings, i.e., the difference betweenthe largest and smallest values, then

rn

.

This is usually adequate for n less than about 12.

Combination of errors

IfZ = Z(A, B, . . .) (with A, B, etc. independent) then

(Z)2 =

Z

AA

2+

Z

BB

2+

So if(i) Z = A B C, (Z)2 = (A)2 + (B)2 + (C)2

(ii) Z = AB or A/B,Z

Z

2=A

A

2+B

B

2(iii) Z = Am,

Z

Z= m

A

A

(iv) Z = ln A, Z =A

A

(v) Z = exp A,Z

Z= A

25


28/28

18. Statistics

Mean and Variance

A random variable X has a distribution over some subset x of the real numbers. When the distribution of X isdiscrete, the probability that X = xi is Pi. When the distribution is continuous, the probability that X lies in aninterval x is f(x)x, where f(x) is the probability density function.

Mean = E(X) = Pixi orZ

x f(x) dx.

Variance2 = V(X) = E[(X )2] = Pi(xi )2 orZ

(x )2f(x) dx.

Probability distributions

Error function: erf(x) =2

Zx0

ey2

dy

Binomial: f(x) =

n

x

pxqnx where q = (1 p), = np, 2 = npq, p < 1.

Poisson: f(x) =

x

x! e

, and 2

=

Normal: f(x) =1

2exp

(x )

2

22

Weighted sums of random variables

IfW = aX+ bYthen E(W) = aE(X) + bE(Y). IfXand Yare independent then V(W) = a2V(X) + b2V(Y).

Statistics of a data sample x1, . . . , xn

Sample mean x =1

n xi

Sample variance s2 =1

n(xi x)2 =

1n

x2i

x2 = E(x2) [E(x)]2

Regression (least squares fitting)

To fit a straight line by least squares to n pairs of points (xi, yi), model the observations by yi = + (xi x) + i,where the i are independent samples of a random variable with zero mean and variance

2.

Sample statistics: s2x =1

n(xi x)2, s2y =

1

n(yi y)2, s2xy =

1

n(xi x)(yi y).

Estimators: = y, = s2xy

s2x ; E(Yat x) = +(x x); 2 = nn 2 (residual variance),where residual variance =

1

n{yi (xi x)}2 = s2y s4xys2x .

Estimates for the variances ofand are 2n

and2

ns2x.

Correlation coefficient: = r = s2xysxsy

.

Mathematical Formula Handbook[1]

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