OTLN2012

SMITTY’S OUTLINE OF APPLIED

MATHEMATICS STUDY BOOKLET

PREPARED by KEITH SMITH 2010

What is MATHEMATICS?? MATHEMATICS is the branch of the natural sciences that deals with sequences, limits, limits of sequences, and continuity. Arithmetic is the study and application of mathematical operations… Throughout history, mathematics has been called “the queen of the sciences…” The origin of this reference is obscure, but I do believe De Cartes was the one who first made this reference…

Many of the statistical equations used in these algorithms are from “Schaum’s Outline of Probability and Statistics”…

ITERATIVE SYSTEM

OUTPUT SIGNAL SYSTEM

FEEDBACK SIGNAL

INPUT SIGNAL

+

-

+

-

USEFUL PROPERTIES OF ANTILOGARITHMS

( )

( )

1)

)

)

ln)

)

0

ln

=

=

=

=

=

−

++

βββ

β

ββββ

βββ

β β

v

iv

iiidxdii

ei

b

aba

cbacba

xx

xx

USEFUL PROPERTIES OF LOGARITHMS

( ) ( )( )

( ) ( )( ) ( ) ( ) ( )

( ) ( )

( ) 01log)

logloglog)

loglogloglog)ln

11log)

lnlnlog)

=

−=

++=

=

=

β

βββ

ββββ

β

β

β

β

v

yxyxiv

zyxxyziiix

xdxdii

xxi

The Riemann-Trapezoidal-Antiderivative generates the natural logarithm function…

( )

( ) τττ

ττ

τ∆

+=

=

∑

∫

+→∆ j jj

x

x

dx

1121limln

1ln

10

1

NOTES FROM INTEGRAL CALCULUS…. RIEMANN-TRAPEZOIDAL-ANTIDERIVATIVE….

( ) ( ) xffdxxfj

jjx

∆+= ∑∫ +→∆

10 2

1lim

RIEMANN SUM….

( ) ( )∑∫ ∆=→∆

xxfdxxfx

lim0

NOTES FROM DIFFERENTIAL CALCULUS…. CENTRAL DIFFERENCE FORMULA….

( )[ ] ( ) ( )h

hxfhxfxf

dxd

h 2lim0

−−+=

→

FORWARD DIFFERENCE FORMULA….

( )[ ] ( ) ( )h

xfhxfxf

dxd

h

−+=

→lim

0

FROM MATHEWS, JOHN H. (NUMERICAL METHODS FOR…) & PRESS, WILLIAM H. (NUMERICAL RECIPES…)

TAYLOR POLYNOMIAL APPROXIMATION…

( ) ( )( ) ( ) ( )∑

=

−=≈N

k

kk

N xxk

xfxPxf0

00

! ;

WHERE P(x) IS A POLYNOMIAL THAT CAN BE USED TO APPROXIMATE f(x).

LET US TAKE THE DERIVATIVE OF LOGb(x):

( ) ( ) ( )

718262.211limlim

,11,

11loglim

:,1,

loglim2

loglim

2loglog

limlog

21

00

21

21

0

21

000

≈

−+

==

−+

=

−+

=

−+

=

−+

=

−−+=

→→

→

→→→

h

hh

hh

bh

h

bh

b

h

bb

hb

hhebseewe

hhblettingand

hh

getwexlettingnow

hxhx

hhxhx

hhxhx

Xdxd

USING DIFFERENTIAL CALCULUS, WE CAN SHOW:

( )

( ) ( )FORMULADIFFERENCEFORWARDhe

FORMULADIFFERENCECENTRALhhe

hh

h

h

1

0

21

0

1lim

&

11lim

+=

−+

=

→

→

Volume & Area formulae from multivariate calculus:

∫

∫ ∫

−

−

−

−−

−=

−−=

r

rcircle

r

r

yr

yr

sphere

dxxrA

dxdyyxrV

22

222

2

&

222

22

( )

{ } { }{ }{ }

( )

( )

( )

( )

∫ ∫− −

−−=

⇒==⇒=≠

⇒=

−−==

r

r

y

yellipsoid dxdy

by

axrcVOLUME

circlezandbandaellipsezandbora

spherecba

by

axrcyxzellipsoid

ρ

ρ

ψ

ψ2

2

2

22

2

2

2

22

2

00.100.1

0.1,0.1,0.1,,

,

( )

( )

( )

( )

∫ ∫− −

−−=r

r

y

yellipsoid dxdy

by

axrcVOLUME

ρ

ρ

ψ

ψ2

2

2

222

ELEMENTARY FORMULA FROM INTEGRAL CALCULUS

SHAPE VOLUME FORMULA SPHERE

3

34 rπ

CYLINDER

hr 2π CONE

hr 2

31

π

“ORDER OF OPERATIONS”

The “Order of Operations” is the foundation upon which algebra is

built. Solutions to algebra problems cannot be found without using

the “Order of Operations”.

rder of Mathematical Operations.

PARENTHESES () EXPONENTS ^ MULTIPLICATION * DIVISION / ADDITION + SUBTRACTION -

Many solutions are straight forward, yet must be worked out.

Algebra problems cannot be worked without using the “Order of

Operations”. The “Order of Operations” is indeed the foundation of

algebra.

QUANTITATIVE OPERATORS

""""

""""

""

TOEQUALORTHANLESSTOEQUALORTHANGREATER

EQUALSTHANLESS

THANGREATER

≤≥=<>

The associative, commutative, and distributive laws of algebra:

f1*(f2*f3) = (f1*f2)*f3

(associative property)

f1+f2 = f2+f1 (commutative property)

f1*(f2+f3) = f1*f2 + f1*f3

(distributive property)

FROM SCHAUM’S OUTLINE OF PROBABILITY & STATISTICS:

“The convolution of two functions f1 & f2, abbreviated as f1*f2, is defined as follows by a function g(u):

( ) ( )∫∞

∞−

−= dxxufxfug 21)(

Note: f1 and f2 obey the associative, commutative, and distributive laws of algebra with respect to convolution.”

ELEMENTS FROM COMBINATORIAL ANALYSIS

The number of combinations of r items selected from a total of n

items is expressed as

( )!!!

rnrnCrn −

=

The number of permutations of n items taken r at a time is

expressed as

( )!!rn

nPrn −=

FROM COMPLEX VARIABLES, WE HAVE:

( ) ( )θθθ sincos iei +=

which is known commonly as EULER’S IDENTITY.

FROM EULER’S IDENTITY, WE HAVE THE FOLLOWING RESULTS:

1

1

1

1

2

2

2

2

=

=

−=

=

−=

−=

−

−

−

i

i

i

i

i

i

e

e

ie

ie

e

e

π

π

π

π

π

π

CONJECTURES FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS

( ) ( )

DAMPEDUNDERbDAMPEDCRITICALLYb

DAMPEDOVERb

tetcfdtdfb

dtfda

000

2

2

<=>

=++

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12 14 16

1.0000X''(t)+0.2500X'(t)+1.0000X(t)+0.0000=0.00 (OVER DAMPED)

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10 12 14 16

1.0000X''(t)+0.0000X'(t)+1.0000X(t)+0.0000=0.00 (CRITICALLY DAMPED)

-6

-4

-2

0

2

4

6

0 2 4 6 8 10 12 14 16

1.0000X''(t)-0.2500X'(t)+1.0000X(t)+0.0000=0.00 (UNDER DAMPED)

( )

( )

( ) ( ) 0221

1;

021

)0;(

2

22

2

22

=+−−

=−−

txdtdxt

dtxdt

tLEGENDRE

dtdxt

dtxdt

tLEGENDRE

( ) ( )

( )

( ) ( )

( ) ( )

( ) ( ) 00.100.0

01

)1;(

00.000.1

0

)0;(

0

);(

00

22

22

00

22

22

222

22

==

=−++

==

=++

=−++

txtx

txtdtdxt

dtxdt

tBESSEL

txtx

txtdtdxt

dtxdt

tBESSEL

txntdtdxt

dtxdt

ntBESSEL

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12 14 16

BESSEL(t;0)

BESSEL(t;1)

BESSEL FUNCTIONS

SOME INTERESTING INPUT FUNCTIONS (SIGNALS):

( ) ( ) ( )( )( ) ( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( )( ) ( )( ) ( )( )( )( ) ( ) ( )( )

( )

( )βντντ

τ

ντ

ντ

ντ

ντ

ντ

ντ

β

τ

τωτωβτϕ

τωτωβτϕ

τωτωβτϕ

τωτωβτϕ

τωτωβτϕ

τωτωβτϕ

ln

ln

22

212

16

2221115

221114

2221113

2112

22111

sincos

sinsincoscos

sincoscossin

sincoscossin

cossin

cossin

−−

−

−

−

−

−

−

=

=

+=

+=

=

+=

=

+=

ee

AABABA

BBABABA

AAA

nn

nn

nn

nn

nn

nn

nn

:NOTE

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0 2 4 6 8 10 12 14 16

e(t)=0.1000 sin(10.0000 exp(1.0000 ln(t))) + 1.0000 cos(1.0000 exp(1.0000 ln(t)))

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0 2 4 6 8 10 12 14 16


-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 2 4 6 8 10 12 14 16


-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12 14 16

e(t)=sin(1.0000 cos(1.0000 exp(1.0000 ln(t))))

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12 14 16

e(t)=sin(25.0000 cos(0.5000 exp(1.0000 ln(t))))

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12 14 16

e(t)=sin(17.0000 cos(0.7000 exp(1.0000 ln(t))))

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12 14 16

e(t)=sin(10.0000 cos(1.0000 exp(1.0000 ln(t))))

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12 14 16 18

e(t)=exp(-0.3700 exp(1.0000 ln(t)))cos(3.7000 exp(1.0000 ln(t)))

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10 12 14 16 18

e(t)=exp(-0.3700 exp(1.0000 ln(t))) (0.1000 sin(10.0000 exp(1.0000 ln(t))) + 1.0000 cos(3.7000 exp(1.0000 ln(t))))

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10 12 14 16 18


-1.0

-0.5

0.0

0.5

1.0

1.5

0 2 4 6 8 10 12 14 16 18


-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 2 4 6 8 10 12 14 16 18


-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 2 4 6 8 10 12 14 16 18

e(t)=exp(-0.3700 exp(1.0000 ln(t))) (sin(10.0000 exp(1.0000 ln(t))) + cos(3.7000 exp(1.0000 ln(t))))

SOME NOTES FROM SIGNAL ANALYSIS: (DECIBEL METER = SIGNAL STRENGTH METER) (DECIBEL IS A MEASURE OF SIGNAL STRENGTH)

The following differential equation, discovered by Karl Pearson, generates probability density functions:

( )( )[ ]

2

1cxbxa

xddx

xfdxf ++

−=⋅

When a=c=0, b>0, & d>-b, the solution is the gamma distribution. When a=c=d=0 & b>0, the solution is the exponential distribution.

When a=0, b=-c, 1&,11−><

−bd

bd

, the solution is the beta

distribution.

Problem 6.30, from FREUND : “Karl Pearson, one of the founders of modern statistics, showed that the differential equation

( )( )[ ]

2

1cxbxa

xddx

xfdxf ++

−=⋅

yields (for appropriate values of the constants a,b,c, & d) most of the important distributions of statistics. Verify that the differential equation gives

a) the gamma distribution when a=c=0, b > 0, & d >-b ; b) the exponential distribution when a=c=d=0 & b > 0 ; c) the beta distribution when a=0, b=-c, 1&,11

−><−

bd

bd . “

The differential equation discovered by Karl Pearson indeed closes the gap between calculus of real variables and random variables… We can therefore think of the equation as an ordinary differential equation of a stochastic variable versus a real variable…

ν )(2 Xνχ

4.0000 ( ) ( ) 2ln2ln2 xxe −+−

5.0000 ( ) ( ) 2ln5.12ln5.2)329340.1ln( xxe −+−−

6.0000 ( ) ( ) 2ln22ln3)000000.2ln( xxe −+−−

7.0000 ( ) ( ) 2ln5.22ln5.3)323351.3ln( xxe −+−−

8.0000 ( ) ( ) 2ln32ln4)000000.6ln( xxe −+−−

9.0000 ( ) ( ) 2ln5.32ln5.4)631728.11ln( xxe −+−−

10.0000 ( ) ( ) 2ln42ln5)000000.24ln( xxe −+−−

11.0000 ( ) ( ) ( ) 2ln5.42ln5.5342777.52ln xxe −+−−

12.0000 ( ) ( ) ( ) 2ln52ln6000000.120ln xxe −+−−

13.0000 ( ) ( ) ( ) 2ln5.52ln5.6885263.287ln xxe −+−−

14.0000 ( ) ( ) ( ) 2ln62ln7000000.720ln xxe −+−−

15.0000 ( ) ( ) ( ) 2ln5.62ln5.7253834.1871ln xxe −+−−

16.0000 ( ) ( ) ( ) 2ln72ln8000000.5040ln xxe −+−−

17.0000 ( ) ( ) ( ) 2ln5.72ln5.8392545.14034ln xxe −+−−

18.0000 ( ) ( ) ( ) 2ln82ln9000000.40320ln xxe −+−−

19.0000 ( ) ( ) ( ) 2ln5.82ln5.9000352.119292ln xxe −+−−

20.0000 ( ) ( ) ( ) 2ln92ln10000000.362880ln xxe −+−−

21.0000 ( ) ( ) ( ) 2ln5.92ln5.10915023.1133263ln xxe −+−−

22.0000 ( ) ( ) ( ) 2ln102ln11000000.3628800ln xxe −+−−

23.0000 ( ) ( ) ( ) 2ln5.102ln5.11458270.11898968ln xxe −+−−

24.0000 ( ) ( ) ( ) 2ln112ln12000000.39916800ln xxe −+−−

25.0000 ( ) ( ) ( ) 2ln5.112ln5.12788888.136829057ln xxe −+−−

p(X) E(X) Var(X)

ννχ

λλλσµσµ

αβαββα

ν 2)();(

),;(),;(

2

22

2

XXpoisson

XNXΓ

DEFINITIONS (converted).

( )( ) ( )( ) ( ) ( ) ( ) .0

.,),;(

.0),;(

2ln122ln22ln2

)()2ln(21

)ln()1()ln())(ln(

2

22

≥∀=

∞∞−∈∀=

≥∀=Γ

−−+−Γ−

−+−

−−+−Γ−

xeX

xeXN

xeX

xx

x

xx

ννν

ν

σµπσ

βαβαα

χ

σµ

βα

THIS PAGE INTENTIONALLY LEFT BLANK

1 5 9

13 17 21 25 29 33 37 41 45 49 53 57 61S1

S10

S19

S28S37

S46S55

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

BVN((X,Y);(0.0000,0.0000,1.0000²,1.0000²);0.0000)

111

21

31

41

51

61

71

81

S1 S6 S11 S1

6 S21 S2

6 S31 S3

6 S41 S4

6

0

0.5

1

1.5

2

2.5

3

3.5

4

z(m,s) = N(m,s²) . s² = [0.10²,0.99²] .

1 7

13 19 25 31 37 43 49 55 61 67 73 79 85 91 97S1

S16

S31

S46S61

S76S91

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

z(x,y)=exp(-5.0000 exp(1.5000 ln(R)))cos(25.0000 exp(1.5000 ln(R)))

1 7

13 19 25 31 37 43 49 55 61 67 73 79 85 91 97S1

S16

S31

S46S61

S76S91

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

z(x,y)=exp(-5.0000 exp(1.7000 ln(R)))cos(25.0000 exp(1.7000 ln(R)))

-1.3

-0.8

-0.3

0.2

0.7

1.2

1.7

2.2

2.7

3.2

3.7

z(x,y)=cos(x)cos(y) [x,y]=[-3,3]

( ) ( )

( )

( ) ( )

( ) ( )

( ) ( ) 00.100.0

01

)1;(

00.000.1

0

)0;(

0

);(

00

22

22

00

22

22

222

22

==

=−++

==

=++

=−++

txtx

txtdtdxt

dtxdt

tBESSEL

txtx

txtdtdxt

dtxdt

tBESSEL

txntdtdxt

dtxdt

ntBESSEL

FROM PRESS, WILLIAM H.:

( )

( )

( )

( )yxyu

xu

ellpticEQUATIONPOISSON

xuD

xtu

parabolicEQUATIONDIFFUSION

xuv

tu

hyperbolicEQUATIONWAVE

,

:

:

:

2

2

2

2

2

22

2

2

ρ=∂∂

+∂∂

∂∂

∂∂

=∂∂

∂∂

=∂∂

where

“v=constant is the velocity of wave propagation

D is the diffusion

coefficient

ρ is the source term “

DEFINITIONS.

( )

( ) ( )( ) ( ) ( ) ( )

( )

.0!

);(

.0)00.2,2;()(

.0

22

.0

.0),;(

.0),;(

2

2122

2ln122ln22ln2

)ln()1()ln())(ln(

1

≥∀=

≥∀==Γ=

≥∀

Γ

=

≥∀=

≥∀=Γ

≥∀Γ

=Γ

−

−−

−−+−Γ−

−−+−Γ−

−−

xx

eXpoisson

xXX

xexX

xeX

xeX

xexX

x

x

xx

xx

x

λλ

βναχ

νχ

χ

βα

αββα

λ

ν

ν

ν

ν

ννν

ν

βαβαα

α

βα

DEFINITIONS.

( ) ( )( )

( ) ( )( )

( )( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( )( )

.0!

);(

.0)00.2,2;()(

.02

.0

.0),;(

.0),;(

20

ln12

2122

2ln122ln2ln2

)ln()1()ln()ln(

0

ln1

1

0

ln12

0

ln1

≥∀=

≥∀==Γ=

≥∀∫

=

≥∀=

≥∀=Γ

≥∀=Γ

−

∞ −−

−−

−−+−

∫−

−−+−∫−

∞−−

−−

+

∞

+

−−

∞

+

−−

+∫

xx

eXpoisson

xXX

xde

exX

xeX

xeX

xde

exX

x

x

xxde

xxde

a

x

a

λλ

βναχ

τχ

χ

βα

τββα

λ

ν

ττνν

ν

ν

νντ

ν

βαβατ

ττα

βα

ττν

ττ

A random variable X has an exponential distribution, and is referred to as an exponential random variable, if and only if its probability density is given by:

( ).00 >>

= −

αα α

andxforexf x

Student’s t distribution If a random variable has the density function

( )( )

( )∞∞−∈∀

+

Γ

+

Γ=

+−

,1

2

21

21

2

tvt

vv

v

tfv

π

it is said to have student’s t distribution, briefly the t distribution, with v degrees of freedom. The forms expressed in Schaum’s Outline of Probability and Statistics and Freund’s text are as expressed above, and identical.

The F distribution: From Schaum’s Outline of Probability and Statistics… “A random variable is said to have the F distribution (named for R.A. Fisher) with v1 and v2 degrees of freedom if its density function is given by:

( ) ( ) ( )

.022

2 212

1222

21

21

21

21121

>∀

+

Γ

Γ

+

Γ=

+−−

u

uvvuvvvv

vv

ufvvvvv

”

From Freund’s text, “Mathematical Statistics”, another formula for the F distribution is given by the following expression… “If U and V are independent random variables having Chi-square distributions with v1 and v2 degrees of freedom, then

2

1

vV

vU

F =

is a random variable having an F distribution, namely, a random variable whose probability density is given by

( )( )211

1

21

2

11

22

2

1

21

21

1

22

2vvv

v

fvvf

vv

vv

vv

fg+−

−

+

Γ

Γ

+

Γ=

for f > 0 and g(f)=0 elsewhere…”

From “Schaum’s Outline of Probability and Statistics”…

( )( )

( )

( ) ( ) ( )

.022

2,;

:

.,1

2

21

;

:'

212

1222

21

21

21

21

21

2

21121

>∀

+

Γ

Γ

+

Γ=

∞∞−∈∀

+

Γ

+

Γ=

+−−

+−

u

uvvuvvvv

vv

vvUfdist

bygivenisondistributiFThe

tvt

vv

v

vTtdist

bygivenisondistributitsStudentThe

vvvvv

v

π

Using logarithms, we can convert the classical form of the expressions to a contemporary form. The F distribution expressed here is derived from the formula published in “Schaum’s Outline of Probability and Statistics”.

From the formulas published in Schaum’s Outline:

( )( ) ( ) ( )( )

( )

( )( ) ( ) ( ) ( ) ( ) ( )( )

.0;0;0

:

.0;,

:'

21

2ln

2ln

2lnlnln2lnln5.0

21ln1ln1

2ln2ln5.0

2121122112211

ln2

>>>∀=

>∞∞−∈∀=

+

Γ+

Γ−

Γ−++−−++

+

Γ+

+++

Γ+− −

vvueuf

bygivenisondistributiFThe

vtetf

bygivenisondistributitsStudentThe

vvvvuvvvvuvvvvv

vetvvv vπ

DEFINITIONS (primary). ( )

( )

( )

( )

( )

( )( )

( )

( ) ( ) ( )

( )

( )( )

( )Freundu

fvvf

vv

vv

vv

Fg

sSchaumu

uvvuvvvv

vv

Uf

tvt

vv

v

Ttdist

andxeXedist

xx

eXpoisson

xexX

xexX

xeXN

vvvv

vvvvv

v

x

x

x

x

x

.0;0;0

1

22

2,;

'.0;0;022

2,;

,1

2

21

;

.00;

.0!

);(

.0

22

.0),;(

.,2

1),;(

21

21

2

11

22

2

1

21

21

21

21

212

1222

21

21

21

21

21

2

2122

1

22

2111

21121

2

2

>>>∀

+

Γ

Γ

+

Γ=

>>>∀

+

Γ

Γ

+

Γ=

∞∞−∈∀

+

Γ

+

Γ=

>>∀=

≥∀=

≥∀

Γ

=

≥∀Γ

=Γ

∞∞−∈∀=

+−−

+−−

+−

−

−

−−

−−

−−

νν

νν

νν

νν

πν

ααα

λλ

νχ

αββα

πσσµ

α

λ

ν

ν

ν

α

βα

σµ

DEFINITIONS (converted).

( )

( ) ( )( ) ( ) ( ) ( )

( )( ) ( ) ( )( )

( )

( )( ) ( ) ( ) ( ) ( ) ( )( )

.0;0;0,;

.0;,;

.0

.0),;(

.,),;(

21

2ln

2ln

2lnlnln2lnln5.0

21

21ln1ln1

2ln2ln5.0

2ln122ln22ln2

)ln()1()ln())(ln(

)()2ln(21

2121122112211

ln2

2

22

>>>∀=

>∞∞−∈∀=

≥∀=

≥∀=Γ

∞∞−∈∀=

+

Γ+

Γ−

Γ−++−−++

+

Γ+

+++

Γ+−

−−+−Γ−

−−+−Γ−

−+−

−

vvueUfdist

vteTtdist

xeX

xeX

xeXN

vvvvuvvvvuvvvvv

vetvvv

xx

xx

x

v

νν

ν

χ

βα

σµ

π

ννν

ν

βαβαα

σµπσ

PROPERTIES OF PROBABILITY DENSITY FUNCTIONS

1)(lim

0)(lim

)()(

0)(

=

=

=

ℜ∈∀≥

∞→

∞→

∞−∫

xP

xp

dpxP

xxp

x

x

x

ττ

p(X) E(X) Var(X)

( )

( )

( ) ( )( )( )2

221

2122

2

221

2

2

2

22

2422

2,;

20;

11;

);(2)(

),;(),;(

−−

−+−

−

Γ

νννννν

νν

νν

ννν

ααα

λλλννχ

αβαββα

σµσµ

ν

Ffdist

Ttdist

Xedist

XpoissonX

XXN

* From Schaum’s Outline of Probability and Statistics, second edition.

Moment Generating Functions NDF(X;m=25.00,s={1.0,2.0,3.0,4.0,5.0} )

0

10

20

30

40

50

60

70

80

90

100

0 0.00000002 0.00000004 0.00000006 0.00000008 0.0000001 0.00000012

DEFINITIONS

∫∞

∞−

= dxxxpXE )()(

( ) ∫∞

∞−

== dxxpeeEtM xtXtX )()(

VARIANCE

)(:]))([()]([)()(

2

222

XVarnoteXEXEXEXEXVar

=

−=−=

σ

STATISTICAL EXPECTATION ),;( 2σµXN

µ σ E(X) Var(X) StdDev(X) 32.000 2.000 32.000 4.000 2.000 32.000 4.000 32.000 16.000 4.000 32.000 6.000 32.000 36.000 6.000 32.000 8.000 32.000 63.996 8.000 32.000 10.000 32.002 99.873 9.994 85.000 5.000 85.000 25.001 5.000 85.000 10.000 85.000 100.004 10.000 85.000 15.000 85.000 225.005 15.000 85.000 20.000 85.000 399.988 20.000 85.000 25.000 85.002 624.608 24.992 170.000 10.000 170.000 100.005 10.000 170.000 20.000 170.000 400.016 20.000 170.000 30.000 170.000 899.990 30.000 170.000 40.000 170.000 1599.965 40.000 170.000 50.000 170.004 2498.445 49.984 320.000 20.000 320.000 400.090 20.002 320.000 40.000 320.000 1600.059 40.001 320.000 60.000 320.000 3599.930 59.999 320.000 80.000 320.001 6399.684 79.998 320.000 100.000 320.018 9987.305 99.937 850.000 50.000 850.000 2500.789 50.008 850.000 100.000 850.000 9999.396 99.997 850.000 150.000 850.000 22499.852 150.000 850.000 200.000 849.999 40000.223 200.001 850.000 250.000 850.021 62459.703 249.919 1700.000 100.000 1700.000 10001.160 100.006 1700.000 200.000 1699.997 40008.789 200.022 1700.000 300.000 1699.998 90008.062 300.013 1700.000 400.000 1699.998 160001.297 400.002 1700.000 500.000 1700.038 249851.500 499.851 3200.000 200.000 3199.995 40004.688 200.012 3200.000 400.000 3199.992 160024.000 400.030 3200.000 600.000 3199.988 360032.000 600.027 3200.000 800.000 3199.987 640029.250 800.018 3200.000 1000.000 3200.169 998746.188 999.373

STATISTICAL EXPECTATION ),;( βαXΓ

α β E(X) Var(X) StdDev(X) 2.000 3.333 6.666 22.222 4.714 4.000 3.333 13.332 44.436 6.666 6.000 3.333 19.998 66.654 8.164 8.000 3.333 26.664 88.868 9.427 10.000 3.333 33.329 111.053 10.538 2.000 8.500 17.000 144.499 12.021 4.000 8.500 34.000 289.000 17.000 6.000 8.500 51.000 433.499 20.821 8.000 8.500 68.000 577.965 24.041 10.000 8.500 84.998 722.193 26.874 2.000 17.000 34.000 577.996 24.042 4.000 17.000 68.000 1156.004 34.000 6.000 17.000 102.000 1734.010 41.641 8.000 17.000 136.000 2311.914 48.082 10.000 17.000 169.996 2888.743 53.747 2.000 33.333 66.666 2222.170 47.140 4.000 33.333 133.332 4444.409 66.666 6.000 33.333 199.998 6666.629 81.649 8.000 33.333 266.663 8888.510 94.279 10.000 33.333 333.325 11107.218 105.391 2.000 85.000 170.001 14449.882 120.208 4.000 85.000 339.999 28899.727 169.999 6.000 85.000 509.999 43349.215 208.205 8.000 85.000 679.997 57798.684 240.414 10.000 85.000 849.977 72221.602 268.741 2.000 170.000 340.001 57799.301 240.415 4.000 170.000 679.995 115601.680 340.002 6.000 170.000 1019.996 173400.219 416.414 8.000 170.000 1359.989 231211.500 480.845 10.000 170.000 1699.954 288881.062 537.477 2.000 333.333 666.665 222220.672 471.403 4.000 333.333 1333.316 444449.844 666.671 6.000 333.333 1999.980 666653.062 816.488 8.000 333.333 2666.630 888998.062 942.867 10.000 333.333 3333.276 1110858.250 1053.973

),00.25;( 2σµ =XN X σ=1.00 σ=2.00 σ=3.00 σ=4.00 σ=5.00

0.001000 0.000000 0.000000 0.000000 0.000000 0.000000 1.001000 0.000000 0.000000 0.000000 0.000000 0.000001 2.001000 0.000000 0.000000 0.000000 0.000000 0.000002 3.001000 0.000000 0.000000 0.000000 0.000000 0.000005 4.001000 0.000000 0.000000 0.000000 0.000000 0.000012 5.001000 0.000000 0.000000 0.000000 0.000000 0.000027 6.001000 0.000000 0.000000 0.000000 0.000001 0.000058 7.001000 0.000000 0.000000 0.000000 0.000004 0.000122 8.000999 0.000000 0.000000 0.000000 0.000012 0.000247 9.000999 0.000000 0.000000 0.000000 0.000033 0.000477 10.000999 0.000000 0.000000 0.000000 0.000088 0.000887 11.000999 0.000000 0.000000 0.000002 0.000218 0.001584 12.000999 0.000000 0.000000 0.000011 0.000508 0.002718 13.000999 0.000000 0.000000 0.000045 0.001109 0.004481 14.000999 0.000000 0.000000 0.000160 0.002275 0.007098 15.000999 0.000000 0.000001 0.000515 0.004385 0.010803 16.000999 0.000000 0.000008 0.001479 0.007939 0.015796 17.000999 0.000000 0.000067 0.003802 0.013504 0.022191 18.000999 0.000000 0.000437 0.008747 0.021579 0.029954 19.000999 0.000000 0.002219 0.018009 0.032392 0.038847 20.000999 0.000001 0.008775 0.033177 0.045677 0.048404 21.000999 0.000134 0.027022 0.054694 0.060508 0.057948 22.000999 0.004445 0.064807 0.080684 0.075298 0.066653 23.000999 0.054099 0.121046 0.106506 0.088027 0.073660 24.000999 0.242213 0.176077 0.125808 0.096673 0.078212 25.000999 0.398942 0.199471 0.132981 0.099736 0.079788 26.000999 0.241729 0.175989 0.125780 0.096661 0.078205 27.000999 0.053883 0.120925 0.106459 0.088005 0.073648 28.000999 0.004419 0.064710 0.080630 0.075270 0.066637 29.000999 0.000133 0.026969 0.054646 0.060478 0.057929 30.000999 0.000001 0.008753 0.033141 0.045648 0.048384 31.000999 0.000000 0.002213 0.017985 0.032367 0.038828 32.000999 0.000000 0.000436 0.008734 0.021560 0.029937 33.000999 0.000000 0.000067 0.003795 0.013491 0.022177 34.000999 0.000000 0.000008 0.001476 0.007930 0.015784 35.000999 0.000000 0.000001 0.000514 0.004379 0.010794 36.000999 0.000000 0.000000 0.000160 0.002272 0.007092 37.000999 0.000000 0.000000 0.000045 0.001107 0.004477 38.000999 0.000000 0.000000 0.000011 0.000507 0.002715 39.000999 0.000000 0.000000 0.000002 0.000218 0.001582 40.000999 0.000000 0.000000 0.000000 0.000088 0.000886 41.000999 0.000000 0.000000 0.000000 0.000033 0.000477 42.000999 0.000000 0.000000 0.000000 0.000012 0.000246 43.000999 0.000000 0.000000 0.000000 0.000004 0.000122 44.000999 0.000000 0.000000 0.000000 0.000001 0.000058 45.000999 0.000000 0.000000 0.000000 0.000000 0.000027 46.000999 0.000000 0.000000 0.000000 0.000000 0.000012 47.000999 0.000000 0.000000 0.000000 0.000000 0.000005 48.000999 0.000000 0.000000 0.000000 0.000000 0.000002 49.000999 0.000000 0.000000 0.000000 0.000000 0.000001

)(2 Xνχ

X ν=5.00 ν=10.00 ν=15.00 ν=20.00 ν=25.00 0.001000 0.000004 0.000000 0.000000 0.000000 0.000000 1.001000 0.080738 0.000793 0.000002 0.000000 0.000000 2.001000 0.138404 0.007676 0.000099 0.000001 0.000000 3.001000 0.154180 0.023553 0.000833 0.000012 0.000000 4.001000 0.143958 0.045134 0.003277 0.000096 0.000001 5.001000 0.122017 0.066821 0.008473 0.000432 0.000011 6.001000 0.097280 0.084030 0.016807 0.001352 0.000056 7.001000 0.074350 0.094413 0.027761 0.003282 0.000200 8.000999 0.055095 0.097683 0.040104 0.006620 0.000562 9.000999 0.039873 0.094899 0.052299 0.011588 0.001321 10.000999 0.028325 0.087725 0.062912 0.018140 0.002690 11.000999 0.019820 0.077899 0.070895 0.025941 0.004882 12.000999 0.013697 0.066915 0.075696 0.034428 0.008053 13.000999 0.009367 0.055900 0.077243 0.042914 0.012262 14.000999 0.006350 0.045603 0.075840 0.050710 0.017438 15.000999 0.004271 0.036450 0.072027 0.057226 0.023383 16.000999 0.002854 0.028619 0.066455 0.062043 0.029790 17.000999 0.001896 0.022122 0.059773 0.064937 0.036281 18.000999 0.001253 0.016864 0.052566 0.065878 0.042461 19.000999 0.000824 0.012698 0.045308 0.065000 0.047958 20.000999 0.000540 0.009455 0.038355 0.062552 0.052466 21.000999 0.000352 0.006971 0.031944 0.058856 0.055770 22.000999 0.000229 0.005093 0.026216 0.054258 0.057753 23.000999 0.000149 0.003690 0.021227 0.049097 0.058402 24.000999 0.000096 0.002653 0.016978 0.043677 0.057787 25.000999 0.000062 0.001895 0.013427 0.038252 0.056048 26.000999 0.000040 0.001344 0.010508 0.033022 0.053369 27.000999 0.000026 0.000948 0.008146 0.028130 0.049960 28.000999 0.000016 0.000665 0.006258 0.023668 0.046037 29.000999 0.000010 0.000464 0.004768 0.019687 0.041803 30.000999 0.000007 0.000323 0.003605 0.016200 0.037443 31.000999 0.000004 0.000223 0.002706 0.013199 0.033112 32.000999 0.000003 0.000154 0.002017 0.010653 0.028933 33.000999 0.000002 0.000105 0.001494 0.008523 0.025000 34.000999 0.000001 0.000072 0.001101 0.006763 0.021374 35.000999 0.000001 0.000049 0.000806 0.005325 0.018093 36.000999 0.000000 0.000033 0.000587 0.004162 0.015172 37.000999 0.000000 0.000023 0.000425 0.003230 0.012611 38.000999 0.000000 0.000015 0.000307 0.002490 0.010394 39.000999 0.000000 0.000010 0.000220 0.001908 0.008499 40.000999 0.000000 0.000007 0.000158 0.001454 0.006897 41.000999 0.000000 0.000005 0.000112 0.001101 0.005557 42.000999 0.000000 0.000003 0.000080 0.000830 0.004447 43.000999 0.000000 0.000002 0.000056 0.000622 0.003535 44.000999 0.000000 0.000001 0.000040 0.000464 0.002793 45.000999 0.000000 0.000001 0.000028 0.000344 0.002194 46.000999 0.000000 0.000001 0.000019 0.000255 0.001713 47.000999 0.000000 0.000000 0.000014 0.000187 0.001331 48.000999 0.000000 0.000000 0.000009 0.000137 0.001028 49.000999 0.000000 0.000000 0.000007 0.000100 0.000790

);( λXpoisson X λ=5.00 λ=7.00 λ=9.00 λ=11.00 λ=13.00

0.001000 0.006786 0.000919 0.000124 0.000017 0.000002 1.001000 0.033730 0.006393 0.001113 0.000184 0.000029 2.001000 0.084282 0.022364 0.005004 0.001012 0.000191 3.001000 0.140423 0.052165 0.015008 0.003709 0.000829 4.001000 0.175485 0.091266 0.033760 0.010198 0.002693 5.001000 0.175450 0.127747 0.060757 0.022431 0.007000 6.001000 0.146184 0.149014 0.091120 0.041116 0.015164 7.001000 0.104403 0.148993 0.117138 0.064602 0.028157 8.000999 0.065244 0.130352 0.131763 0.088817 0.045749 9.000999 0.036243 0.101374 0.131749 0.108542 0.066075 10.000999 0.018120 0.070956 0.118564 0.119386 0.085890 11.000999 0.008236 0.045152 0.097002 0.119380 0.101502 12.000999 0.003432 0.026339 0.072753 0.109434 0.109963 13.000999 0.001320 0.014185 0.050376 0.092613 0.109980 14.000999 0.000472 0.007096 0.032399 0.072800 0.102170 15.000999 0.000157 0.003314 0.019458 0.053438 0.088633 16.000999 0.000049 0.001453 0.010966 0.036807 0.072149 17.000999 0.000015 0.000600 0.005825 0.023897 0.055358 18.000999 0.000004 0.000235 0.002929 0.014687 0.040208 19.000999 0.000001 0.000087 0.001400 0.008580 0.027761 20.000999 0.000000 0.000031 0.000639 0.004784 0.018295 21.000999 0.000000 0.000011 0.000279 0.002557 0.011554 22.000999 0.000000 0.000003 0.000117 0.001315 0.007021 23.000999 0.000000 0.000001 0.000048 0.000653 0.004121 24.000999 0.000000 0.000000 0.000019 0.000314 0.002345 25.000999 0.000000 0.000000 0.000007 0.000147 0.001298 26.000999 0.000000 0.000000 0.000003 0.000067 0.000701 27.000999 0.000000 0.000000 0.000001 0.000030 0.000371 28.000999 0.000000 0.000000 0.000000 0.000013 0.000192 29.000999 0.000000 0.000000 0.000000 0.000006 0.000098 30.000999 0.000000 0.000000 0.000000 0.000002 0.000049 31.000999 0.000000 0.000000 0.000000 0.000001 0.000025 32.000999 0.000000 0.000000 0.000000 0.000000 0.000012 33.000999 0.000000 0.000000 0.000000 0.000000 0.000006 34.000999 0.000000 0.000000 0.000000 0.000000 0.000003 35.000999 0.000000 0.000000 0.000000 0.000000 0.000001 36.000999 0.000000 0.000000 0.000000 0.000000 0.000001 37.000999 0.000000 0.000000 0.000000 0.000000 0.000000 38.000999 0.000000 0.000000 0.000000 0.000000 0.000000 39.000999 0.000000 0.000000 0.000000 0.000000 0.000000 40.000999 0.000000 0.000000 0.000000 0.000000 0.000000 41.000999 0.000000 0.000000 0.000000 0.000000 0.000000 42.000999 0.000000 0.000000 0.000000 0.000000 0.000000 43.000999 0.000000 0.000000 0.000000 0.000000 0.000000 44.000999 0.000000 0.000000 0.000000 0.000000 0.000000 45.000999 0.000000 0.000000 0.000000 0.000000 0.000000 46.000999 0.000000 0.000000 0.000000 0.000000 0.000000 47.000999 0.000000 0.000000 0.000000 0.000000 0.000000 48.000999 0.000000 0.000000 0.000000 0.000000 0.000000 49.000999 0.000000 0.000000 0.000000 0.000000 0.000000

Notes from 12/19/2011... From debugging, and rerunning the "CALC2011" program, We have observed and confirmed from the textbook by Freund, that Moment Generating Functions can only exist for terminal values in a neighborhood`of zero... This is absolutely fascinating, and shows that there are indeed more schools of probability and statistics... These results are important, in that we can comfortably assert that there exists a school of micro-statistics, and a school of macro-statistics... From Freund... page 161: "The independent variable is t, and we are usually interested in values of t in the neighborhhod of 0. "

∫ = dxXN ),00.25;( 2σµ

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60

σ=1.00

σ=2.00

σ=3.00

σ=4.00

σ=5.00

),00.25,( 2σµ =XN

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 10 20 30 40 50 60

σ=1.00

σ=2.00

σ=3.00

σ=4.00 σ=5.00

THIS PAGE INTENTIONALLY LEFT BLANK

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 10 20 30 40 50 60

ν=5.00

ν=10.00

ν=15.00 ν=20.00

ν=25.00

( )X2νχ

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60

ν=5.00

ν=15.00

ν=20.00

ν=25.00

( )∫ dxX2νχ

ν=10.00

AN EXPRESSION OF THE CENTRAL LIMIT THEOREM

FROM HUNTSBERGER/BILLINGSLEY…

“ If n is large, then

n

Xσ

µ−

has approximately a standard normal

distribution, or (what is the same), X has approximately a normal distribution

with mean μ and standard deviation

nσ

. “

A percentage increase/decrease can be calculated as follows…

100×−

OLDOLDNEW

To illustrate, let us use “proof by induction” starting with 5 and 10… (our minds are designed to comprehend and use these values easily…) … A 100% increase from 5 is 10…

155

5510

==−

Multiply by 100 and we get 1*100=100%...

θ\r 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 r/θ 0.1 0.178412 0.100000 0.035683 0.010000 0.002379 0.000500 0.000095 0.000017 0.000003 0.0000004 0.1 0.2 0.252313 0.200000 0.100926 0.040000 0.013457 0.004000 0.001077 0.000267 0.000062 0.000013 0.2 0.3 0.309020 0.300000 0.185412 0.090000 0.037082 0.013500 0.004450 0.001350 0.000381 0.000101 0.3 0.4 0.356825 0.400000 0.285460 0.160000 0.076123 0.032000 0.012180 0.004267 0.001392 0.000427 0.4 0.5 0.398942 0.500000 0.398942 0.250000 0.132981 0.062500 0.026596 0.010417 0.003799 0.001302 0.5 0.6 0.437020 0.600000 0.524423 0.360000 0.209770 0.108000 0.050345 0.021600 0.008631 0.003240 0.6 0.7 0.472035 0.700000 0.660849 0.490000 0.308396 0.171500 0.086351 0.040017 0.017270 0.007003 0.7 0.8 0.472035 0.800000 0.807403 0.640000 0.430614 0.256000 0.137797 0.068267 0.031496 0.013653 0.8 0.9 0.535237 0.900000 0.963427 0.810000 0.578056 0.364500 0.208100 0.109350 0.053511 0.024604 0.9 θ\r 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 r/θ 1.0 0.564190 1.000000 1.128379 1.000000 0.752253 0.500000 0.300901 0.166667 0.085972 0.041667 1.0 2.0 0.797885 2.000000 3.191538 4.000000 4.255385 4.000000 3.404307 2.666667 1.945319 1.333333 2.0 3.0 0.977205 3.000000 5.863229 9.000000 11.726464 13.500000 14.071752 13.500000 12.061502 10.125000 3.0 4.0 1.128379 4.000000 9.027033 16.000000 24.072096 32.000000 38.515342 42.666667 44.017535 42.666667 4.0 5.0 1.261566 5.000000 12.615617 25.000000 42.052221 62.500000 84.104869 104.166667 120.148958 130.208333 5.0 6.0 1.381977 6.000000 16.583717 36.000000 66.333669 108.000000 159.203768 216.000000 272.920756 324.000000 6.0 7.0 1.492705 7.000000 20.897873 49.000000 97.522079 171.500000 273.064747 400.166667 546.131237 700.291667 7.0 8.0 1.595769 8.000000 25.532304 64.000000 136.157793 256.000000 435.764985 682.666667 995.999906 1365.333333 8.0 9.0 1.692569 9.000000 30.466235 81.000000 182.797478 364.500000 658.070724 1093.500000 1692.181935 2460.375000 9.0 10.0 1.784124 10.000000 35.682480 100.000000 237.862398 500.000000 951.539576 1666.666667 2718.667424 4166.666667 10.0 θ\r 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 r/θ

11.0 1.871205 11.000000 41.166510 121.000000 301.879128 665.500000 1328.297854 2440.166667 4174.676368 6710.458333 11.0 12.0 1.954410 12.000000 46.905837 144.000000 375.223795 864.000000 1801.194036 3456.000000 6175.488285 10368.000000 12.0 13.0 2.034214 13.000000 52.889572 169.000000 458.347752 1098.500000 2383.558041 4760.166667 8853.207365 15470.541667 13.0 14.0 2.111004 14.000000 59.108110 196.000000 551.702349 1372.000000 3089.381772 6402.666667 12357.536215 22409.333333 14.0 15.0 2.185096 15.000000 65.552900 225.000000 655.513262 1687.500000 3933.168660 8437.500000 16856.463631 31640.625000 15.0 16.0 2.256758 16.000000 72.216261 256.000000 770.307070 2048.000000 4929.963762 10922.666667 22536.978169 43690.666667 16.0 17.0 2.326213 17.000000 79.091244 289.000000 896.384672 2456.500000 6095.293576 13920.166667 29605.738717 59160.708333 17.0 18.0 2.393654 18.000000 86.171525 324.000000 1033.640754 2916.000000 7445.226219 17496.000000 38289.702098 78732.000000 18.0 19.0 2.459245 19.000000 93.451317 361.000000 1183.745317 3429.500000 8996.251073 21720.166667 48836.776445 103170.791667 19.0 20.0 2.523132 20.000000 100.925292 400.000000 1345.705388 4000.000000 10765.369051 26666.666667 61516.380025 133333.333333 20.0 θ\r 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 r/θ

21.0 2.585441 21.000000 108.588533 441.000000 1520.228083 4630.500000 12770.002326 32413.500000 76620.077430 170170.875000 21.0 22.0 2.646284 22.000000 116.436474 484.000000 1707.764755 5324.000000 15028.054515 39042.666667 94462.146983 214734.666667 22.0 23.0 2.705758 23.000000 124.460212 529.000000 1908.465855 6083.500000 17557.850495 46640.166667 115380.130966 268180.958333 23.0 24.0 2.763953 24.000000 132.697379 576.000000 2122.707509 6912.000000 20378.076225 55296.000000 139735.369 331776.000 24.0 25.0 2.820948 25.000000 141.047384 625.000000 2350.790618 7812.500000 23507.899105 65104.166667 167913.572 406901.042 25.0 26.0 2.876814 26.000000 149.623065 676.000000 2592.940858 8788.000000 26966.877709 76162.666667 200325.309 495057.333 26.0 27.0 2.931615 27.000000 158.311584 729.000000 2849.534355 9841.500000 30774.931688 88573.500000 237406.55 597871.13 27.0 28.0 2.985411 28.000000 167.225778 784.000000 3120.721561 10976.000000 34952.401958 102442.666667 279619.14 717098.67 28.0 29.0 3.038254 29.000000 176.252811 841.000000 3406.878601 12194.500000 39519.990516 117880.166667 327451.32 854631.21 29.0 30.0 3.090194 30.000000 185.392682 900.000000 3708.231152 13500.000000 44498.790528 135000.000000 381418.2 1012500.00 30.0 θ/r 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 r\θ

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CALCBOOK

APPLICATION INPUT SPECIFICATIONS

DESCRIPTION WORKING DIRECTORY

BSSLCALC BSSLMENU GENERATES SOLUTIONS TO BESSEL EQUATIONS C:\SMITTY\WORKBOOK\ CALC2000 MENU2000 GENERATES STATISTICAL TABLES AND DIFFERENCE TABLES C:\SMITTY\WORKBOOK\ CALC2010 MENU2010 GENERATES SOLTUIONS TO ORDINARY DIFFERENTIAL

EQUATIONS C:\SMITTY\WORKBOOK\

CALC2011 MENU2011 GENERATES STATISTICAL TABLES C:\SMITTY\WORKBOOK\ CALC2012 MENU2012 TABULATES LOGARITHMIC FUNCTIONS C:\SMITTY\WORKBOOK\

*The statistical formulae used in these algorithms comes from “Schaum’s Outline of Probability and Statistics”.

NOTES FROM DAVID JEREMIAH, AND OTHERS FROM AFA/AFN (SPRING 2010)

Rev. Ch. 3-4 Rapture occurs between Ch. 3 & Ch. 4 (Lycena)(Not a place, but an age…) -Spirit of GOD has been transferred… Ezekiel 10(14-17)…….[ Four Living Creatures] Ezekiel 1()…………….”” (Angelic Protection) (Execute Judgment) (Lead Worship) 1 COR 15: 1 THESS 4: RAPTURE?? REV 4: REV 21:21 1 COR 15:3-8 “THE CREED” REV 4:6-8 (FOUR LIVING CREATURES) EX 24:10 (SEA OF GLASS) REV 21:21 (TRANSPARENT GLASS) REV 4:5 (SEA OF GLASS) ISRAEL = GOD PREVAILS/RULES IS 11:… “WHERE THE SPIRIT OF THE LORD IS, THERE IS OUR LIBERTY…” (THUNDERING, & LIGHTNING, & VOICES) (16:18)

Mom, thanks for pressing me to settle for nothing less than the best… Dad, let’s go have some coffee…

Sandy, I believe in us, darling… No one loves you the way I do…

The Lord’s Prayer: Matthew 6:9-14 Luke 11:2-4

Our Father, which art in heaven, Hallowed be thy name. Thy kingdom come. Thy will be done in earth as it is in heaven. Give us this day our daily

bread. And forgive us our debts, as we forgive our debtors. And lead us not into temptation, but deliver us from

evil: For thine is the kingdom, and the power, and the glory, for ever.

Amen.

The Priestly Blessing: Numbers 6:24-26

The LORD bless thee, and keep thee:

The LORD make his face

shine upon thee, and be gracious unto thee:

The LORD lift up his

countenance upon thee, and give thee peace.

PSALMS 118:24

This is the day which the LORD hath made; we will rejoice

and be glad in it.

I Am Revelation 1:8

“I am the Alpha and Omega, the beginning and the ending, saith the Lord, which is, and which was, and which is to

come, the Almighty.”

PREPARED AND EDITED

BY

KEITH SMITH 2012

OTLN2012

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