This document provides an outline of applied mathematics concepts including definitions of mathematics, logarithms, integrals, derivatives, differential equations, and transforms. It summarizes key formulas from calculus, statistics, and other areas of mathematics. Examples of functions and signals are shown along with their graphs.

1. SMITTY’S
OUTLINE OF
APPLIED
MATHEMATICS
STUDY BOOKLET
PREPARED by KEITH SMITH
2010

2. 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…

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

4. ITERATIVE SYSTEM
OUTPUT
SIGNALSYSTEM
FEEDBACK
SIGNAL
INPUT
SIGNAL
+
-
+
-

5. USEFUL PROPERTIES OF ANTILOGARITHMS
( )
( )
1)
)
)
ln)
)
0
ln
=
=
=
=
=
−
++
β
β
β
β
ββββ
βββ
β β
v
iv
iii
dx
d
ii
ei
b
a
ba
cbacba
xx
xx

6. USEFUL PROPERTIES OF LOGARITHMS
( ) ( )
( )
( )
( )
( ) ( ) ( ) ( )
( ) ( )
( ) 01log)
logloglog)
loglogloglog)
ln
11
log)
ln
ln
log)
=
−=





++=
=
=
β
βββ
ββββ
β
β
β
β
v
yx
y
x
iv
zyxxyziii
x
x
dx
d
ii
x
xi

7. The Riemann-Trapezoidal-Antiderivative
generates the natural logarithm function…
( )
( ) τ
ττ
τ
τ
τ
∆








+=
=
∑
∫
+
→∆
j jj
x
x
dx
11
2
1
limln
1
ln
1
0
1


8. NOTES FROM INTEGRAL CALCULUS….
RIEMANN-TRAPEZOIDAL-ANTIDERIVATIVE….
( ) ( ) xffdxxf
j
jj
x
∆+= ∑∫ +
→∆
1
0 2
1
lim
RIEMANN SUM….
( ) ( )∑∫ ∆=
→∆
xxfdxxf
x
lim0

9. NOTES FROM DIFFERENTIAL CALCULUS….
CENTRAL DIFFERENCE FORMULA….
( )[ ]
( ) ( )
h
hxfhxf
xf
dx
d
h 2lim0
−−+
=
→
FORWARD DIFFERENCE FORMULA….
( )[ ]
( ) ( )
h
xfhxf
xf
dx
d
h
−+
=
→
lim0

10. FROM MATHEWS, JOHN H. (NUMERICAL METHODS FOR…)
& PRESS, WILLIAM H. (NUMERICAL RECIPES…)
TAYLOR POLYNOMIAL APPROXIMATION…
( ) ( )
( )
( )( )∑=
−=≈
N
k
k
k
N xx
k
xf
xPxf
0
0
0
! ;
WHERE P(x) IS A POLYNOMIAL THAT CAN BE USED TO
APPROXIMATE f(x).

11. LET US TAKE THE DERIVATIVE OF LOGb(x):
( )
( ) ( )
718262.2
1
1
limlim
,
1
1
,
1
1
loglim
:,1,
loglim
2
log
lim
2
loglog
limlog
2
1
00
2
1
2
1
0
2
1
000
≈





−
+
==






−
+
=














−
+
=














−
+
=


















−
+
=




 −−+
=
→→
→
→→→
h
hh
hh
b
h
h
b
h
b
h
bb
h
b
h
h
ebseewe
h
h
blettingand
h
h
getwexlettingnow
hx
hx
h
hx
hx
h
hxhx
X
dx
d

12. USING DIFFERENTIAL CALCULUS, WE CAN SHOW:
( )
( ) ( )FORMULADIFFERENCEFORWARDhe
FORMULADIFFERENCECENTRAL
h
h
e
h
h
h
h
1
0
2
1
0
1lim
&
1
1
lim
+=




−
+
=
→
→

13. Volume & Area formulae from multivariate calculus:
∫
∫ ∫
−
−
−
−−
−=
−−=
r
r
circle
r
r
yr
yr
sphere
dxxrA
dxdyyxrV
22
222
2
&
2
22
22

14. ( )
{ } { }
{ }
{ }
( )
( )
( )
( )
∫ ∫− −
−−=
⇒==
⇒=≠
⇒=
−−==
r
r
y
y
ellipsoid dxdy
b
y
a
x
rcVOLUME
circlezandbanda
ellipsezandbora
spherecba
b
y
a
x
rcyxzellipsoid
ρ
ρ
ψ
ψ
2
2
2
2
2
2
2
2
2
2
2
00.1
00.1
0.1,0.1,0.1,,
,

15. ( )
( )
( )
( )
∫ ∫− −
−−=
r
r
y
y
ellipsoid dxdy
b
y
a
x
rcVOLUME
ρ
ρ
ψ
ψ
2
2
2
2
2
2

16. ELEMENTARY FORMULA FROM INTEGRAL CALCULUS
SHAPE VOLUME FORMULA
SPHERE
3
3
4
rπ
CYLINDER
hr2
π
CONE
hr2
3
1
π

17. “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.

18. QUANTITATIVE OPERATORS
""
""
""
""
""
TOEQUALORTHANLESS
TOEQUALORTHANGREATER
EQUALS
THANLESS
THANGREATER
≤
≥
=
<
>

19. 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)

20. 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.”

21. ELEMENTS FROM COMBINATORIAL ANALYSIS
The number of combinations of r items selected from a total of n
items is expressed as
( )!!
!
rnr
n
Crn
−
=
The number of permutations of n items taken r at a time is
expressed as
( )!
!
rn
n
Prn
−
=

22. FROM COMPLEX VARIABLES, WE HAVE:
( ) ( )θθθ
sincos iei
+=
which is known commonly as EULER’S IDENTITY.

23. 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

24. CONJECTURES FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS
( ) ( )
DAMPEDUNDERb
DAMPEDCRITICALLYb
DAMPEDOVERb
tetcf
dt
df
b
dt
fd
a
0
0
0
2
2
<
=
>
=++

25. -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)

26. -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)

27. -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)

28. ( )
( )
( ) ( ) 0221
1;
021
)0;(
2
2
2
2
2
2
=+−−
=−−
tx
dt
dx
t
dt
xd
t
tLEGENDRE
dt
dx
t
dt
xd
t
tLEGENDRE

29. ( ) ( )
( )
( ) ( )
( ) ( )
( ) ( ) 00.100.0
01
)1;(
00.000.1
0
)0;(
0
);(
00
2
2
2
2
00
2
2
2
2
22
2
2
2
==
=−++
==
=++
=−++
txtx
txt
dt
dx
t
dt
xd
t
tBESSEL
txtx
txt
dt
dx
t
dt
xd
t
tBESSEL
txnt
dt
dx
t
dt
xd
t
ntBESSEL



30. -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

31. SOME INTERESTING INPUT FUNCTIONS (SIGNALS):
( ) ( ) ( )( )
( ) ( ) ( )
( ) ( )( ) ( )( )( )
( ) ( )( ) ( )( )
( ) ( )( ) ( )( )( )
( ) ( ) ( )( )
( )
( )βντντ
τ
ντ
ντ
ντ
ντ
ντ
ντ
β
τ
τωτωβτϕ
τωτωβτϕ
τωτωβτϕ
τωτωβτϕ
τωτωβτϕ
τωτωβτϕ
ln
ln
2
2
21
2
16
2221115
221114
2221113
2112
22111
sincos
sinsincoscos
sincoscossin
sincoscossin
cossin
cossin
−−
−
−
−
−
−
−
=
=
+=
+=
=
+=
=
+=
e
e
AA
BABA
BBA
BABA
A
AA
nn
nn
nn
nn
nn
nn
nn
:NOTE

32. -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)))

33. -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)=0.2500 sin(10.0000 exp(1.0000 ln(t))) + 1.0000 cos(1.0000 exp(1.0000 ln(t)))

34. -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)=0.5000 sin(10.0000 exp(1.0000 ln(t))) + 1.0000 cos(1.0000 exp(1.0000 ln(t)))

35. -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
e(t)=0.7500 sin(10.0000 exp(1.0000 ln(t))) + 1.0000 cos(1.0000 exp(1.0000 ln(t)))

36. -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
e(t)=1.0000 sin(10.0000 exp(1.0000 ln(t))) + 1.0000 cos(1.0000 exp(1.0000 ln(t)))

37. -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))))

38. -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))))

39. -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))))

40. -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))))

41. -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)))

42. -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))))

43. -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))) (0.1700 sin(10.0000 exp(1.0000 ln(t))) + 1.0000 cos(3.7000 exp(1.0000 ln(t))))

44. -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))) (0.2500 sin(10.0000 exp(1.0000 ln(t))) + 1.0000 cos(3.7000 exp(1.0000 ln(t))))

45. -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))) (0.3700 sin(10.0000 exp(1.0000 ln(t))) + 1.0000 cos(3.7000 exp(1.0000 ln(t))))

46. -1.0
-0.5
0.0
0.5
1.0
1.5
0 2 4 6 8 10 12 14 16 18
e(t)=exp(-0.3700 exp(1.0000 ln(t))) (0.5000 sin(10.0000 exp(1.0000 ln(t))) + 1.0000 cos(3.7000 exp(1.0000 ln(t))))

47. -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))) (0.7500 sin(10.0000 exp(1.0000 ln(t))) + 1.0000 cos(3.7000 exp(1.0000 ln(t))))

48. -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))))

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

50. The following differential equation, discovered by
Karl Pearson, generates probability density
functions:
( )
( )[ ]
2
1
cxbxa
xd
dx
xfd
xf ++
−
=⋅
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&,1
1
−><
−
b
d
b
d
, the solution is the beta
distribution.

51. Problem 6.30, from FREUND :
“Karl Pearson, one of the founders of modern statistics, showed that the
differential equation
( )
( )[ ]
2
1
cxbxa
xd
dx
xfd
xf ++
−
=⋅
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&,1
1
−><
−
b
d
b
d
. “

52. 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…

53. ν )(2
Xνχ
4.0000 ( ) ( ) 2
ln2ln2 xx
e
−+−
5.0000 ( ) ( ) 2
ln5.12ln5.2)329340.1ln( xx
e
−+−−
6.0000 ( ) ( ) 2
ln22ln3)000000.2ln( xx
e
−+−−
7.0000 ( ) ( ) 2
ln5.22ln5.3)323351.3ln( xx
e
−+−−
8.0000 ( ) ( ) 2
ln32ln4)000000.6ln( xx
e
−+−−
9.0000 ( ) ( ) 2
ln5.32ln5.4)631728.11ln( xx
e
−+−−
10.0000 ( ) ( ) 2
ln42ln5)000000.24ln( xx
e
−+−−
11.0000 ( ) ( ) ( ) 2
ln5.42ln5.5342777.52ln xx
e
−+−−
12.0000 ( ) ( ) ( ) 2
ln52ln6000000.120ln xx
e
−+−−
13.0000 ( ) ( ) ( ) 2
ln5.52ln5.6885263.287ln xx
e
−+−−
14.0000 ( ) ( ) ( ) 2
ln62ln7000000.720ln xx
e
−+−−
15.0000 ( ) ( ) ( ) 2
ln5.62ln5.7253834.1871ln xx
e
−+−−
16.0000 ( ) ( ) ( ) 2
ln72ln8000000.5040ln xx
e
−+−−
17.0000 ( ) ( ) ( ) 2
ln5.72ln5.8392545.14034ln xx
e
−+−−
18.0000 ( ) ( ) ( ) 2
ln82ln9000000.40320ln xx
e
−+−−
19.0000 ( ) ( ) ( ) 2
ln5.82ln5.9000352.119292ln xx
e
−+−−
20.0000 ( ) ( ) ( ) 2
ln92ln10000000.362880ln xx
e
−+−−
21.0000 ( ) ( ) ( ) 2
ln5.92ln5.10915023.1133263ln xx
e
−+−−
22.0000 ( ) ( ) ( ) 2
ln102ln11000000.3628800ln xx
e
−+−−
23.0000 ( ) ( ) ( ) 2
ln5.102ln5.11458270.11898968ln xx
e
−+−−
24.0000 ( ) ( ) ( ) 2
ln112ln12000000.39916800ln xx
e
−+−−
25.0000 ( ) ( ) ( ) 2
ln5.112ln5.12788888.136829057ln xx
e
−+−−

54. p(X) E(X) Var(X)
ννχ
λλλ
σµσµ
αβαββα
ν 2)(
);(
),;(
),;(
2
22
2
X
Xpoisson
XN
XΓ

55. DEFINITIONS (converted).
( )
( ) ( )( ) ( ) ( ) ( )
.0
.,),;(
.0),;(
2
ln1
2
2ln
22
ln2
)(
)2ln(
2
1
)ln()1()ln())(ln(
2
2
2
≥∀=
∞∞−∈∀=
≥∀=Γ
−−+−Γ−







 −
+−
−−+−Γ−
xeX
xeXN
xeX
xx
x
xx
ννν
ν
σ
µ
πσ
βαβαα
χ
σµ
βα

56. THIS PAGE INTENTIONALLY LEFT BLANK

57. 1
5
9
13
17
21
25
29
33
37
41
45
49
53
57
61
S1
S10
S19
S28
S37
S46
S55
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)

58. 1
11
21
31
41
51
61
71
81
S1
S6
S11
S16
S21
S26
S31
S36
S41
S46
0
0.5
1
1.5
2
2.5
3
3.5
4
z(m,s) = N(m,s²) .
s² = [0.10²,0.99²] .

59. 1
7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97 S1
S16
S31
S46
S61
S76
S91
-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)))

60. 1
7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97 S1
S16
S31
S46
S61
S76
S91
-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)))

61. -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]

62. ( ) ( )
( )
( ) ( )
( ) ( )
( ) ( ) 00.100.0
01
)1;(
00.000.1
0
)0;(
0
);(
00
2
2
2
2
00
2
2
2
2
22
2
2
2
==
=−++
==
=++
=−++
txtx
txt
dt
dx
t
dt
xd
t
tBESSEL
txtx
txt
dt
dx
t
dt
xd
t
tBESSEL
txnt
dt
dx
t
dt
xd
t
ntBESSEL



63. FROM PRESS, WILLIAM H.:
( )
( )
( )
( )yx
y
u
x
u
ellpticEQUATIONPOISSON
x
u
D
xt
u
parabolicEQUATIONDIFFUSION
x
u
v
t
u
hyperbolicEQUATIONWAVE
,
:
:
:
2
2
2
2
2
2
2
2
2
ρ=
∂
∂
+
∂
∂






∂
∂
∂
∂
=
∂
∂
∂
∂
=
∂
∂
where
“v=constant is the velocity
of wave propagation
D is the diffusion
coefficient
ρ is the source term “

64. DEFINITIONS.
( )
( ) ( )( ) ( ) ( ) ( )
( )
.0
!
);(
.0)00.2,2;()(
.0
2
2
.0
.0),;(
.0),;(
2
2
1
2
2
2
ln1
2
2ln
22
ln2
)ln()1()ln())(ln(
1
≥∀=
≥∀==Γ=
≥∀






Γ
=
≥∀=
≥∀=Γ
≥∀
Γ
=Γ
−
−−
−−+−Γ−
−−+−Γ−
−−
x
x
e
Xpoisson
xXX
x
ex
X
xeX
xeX
x
ex
X
x
x
xx
xx
x
λ
λ
βναχ
ν
χ
χ
βα
αβ
βα
λ
ν
ν
ν
ν
ννν
ν
βαβαα
α
βα

65. DEFINITIONS.
( ) ( )( )
( ) ( )( )
( )
( ) ( )( ) ( ) ( ) ( )
( ) ( ) ( )( )
.0
!
);(
.0)00.2,2;()(
.0
2
.0
.0),;(
.0),;(
2
0
ln1
2
2
1
2
2
2
ln1
2
2ln
2
ln
2
)ln()1()ln()ln(
0
ln1
1
0
ln1
2
0
ln1
≥∀=
≥∀==Γ=
≥∀
∫
=
≥∀=
≥∀=Γ
≥∀=Γ
−
∞ −−
−−
−−+−








∫−
−−+−∫−
∞
−−
−−
+
∞
+
−−
∞
+
−−
+
∫
x
x
e
Xpoisson
xXX
x
de
ex
X
xeX
xeX
x
de
ex
X
x
x
xxde
xxde
a
x
a
λ
λ
βναχ
τ
χ
χ
βα
τβ
βα
λ
ν
ττν
ν
ν
ν
νντ
ν
βαβατ
ττα
βα
ττν
ττ

66. 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 >>
= −
α
α α
andxfor
exf x

67. Student’s t distribution
If a random variable has the density function
( )
( )
( )∞∞−∈∀





+






Γ





 +
Γ
=
+−
,1
2
2
1
2
1
2
t
v
t
v
v
v
tf
v
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.

68. 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:

69. ( ) ( )
( )
.0
22
2 2
12
1
22
2
2
1
21
21
21
121
>∀
+






Γ





Γ





 +
Γ
=
+−−





u
uvvuvv
vv
vv
uf
vvvvv
”

70. 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
v
V
v
U
F =
is a random variable having an F distribution, namely, a random
variable whose probability density is given by
( )
( )21
1
1
2
1
2
1
1
2
2
2
1
21
21
1
22
2
vvv
v
f
v
v
f
v
v
vv
vv
fg
+−
−






+











Γ





Γ





 +
Γ
=
for f > 0 and g(f)=0 elsewhere…”

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

72. ( )
( )
( )
( ) ( )
( )
.0
22
2
,;
:
.,1
2
2
1
;
:'
2
12
1
22
2
2
1
21
21
21
2
1
2
21
121
>∀
+






Γ





Γ





 +
Γ
=
∞∞−∈∀





+






Γ





 +
Γ
=
+−−





+−
u
uvvuvv
vv
vv
vvUfdist
bygivenisondistributiFThe
t
v
t
v
v
v
vTtdist
bygivenisondistributitsStudentThe
vvvvv
v
π

73. 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”.

74. From the formulas published in Schaum’s Outline:
( )
( ) ( ) ( )( )
( )
( )
( ) ( ) ( ) ( ) ( ) ( )( )
.0;0;0
:
.0;,
:'
21
2
ln
2
ln
2
lnlnln2lnln5.0
2
1
ln1ln1
2
ln2ln5.0
2121
122112211
ln2
>>>∀
=
>∞∞−∈∀
=











 +
Γ+











Γ−











Γ−++−−++











 +
Γ+





+++











Γ+− −
vvu
euf
bygivenisondistributiFThe
vt
etf
bygivenisondistributitsStudentThe
vvvv
uvvvvuvvvvv
v
etv
v
v v
π

75. DEFINITIONS (primary).
( )
( )
( )
( )
( )
( )
( )
( )
( ) ( )
( )
( )
( )
( )
( )Freundu
f
v
v
f
v
v
vv
vv
Fg
sSchaumu
uvvuvv
vv
vv
Uf
t
v
t
v
v
v
Ttdist
andxeXedist
x
x
e
Xpoisson
x
ex
X
x
ex
X
xeXN
vvv
v
vvvvv
v
x
x
x
x
x
.0;0;0
1
22
2
,;
'.0;0;0
22
2
,;
,1
2
2
1
;
.00;
.0
!
);(
.0
2
2
.0),;(
.,
2
1
),;(
21
2
1
2
1
1
2
2
2
1
21
21
21
21
2
12
1
22
2
2
1
21
21
21
2
1
2
2
1
2
2
1
2
2
21
1
1
21
121
2
2
>>>∀






+











Γ





Γ





 +
Γ
=
>>>∀
+






Γ





Γ





 +
Γ
=
∞∞−∈∀





+






Γ





 +
Γ
=
>>∀=
≥∀=
≥∀






Γ
=
≥∀
Γ
=Γ
∞∞−∈∀=
+−
−
+−−





+−
−
−
−−
−−
−
−
νν
νν
νν
νν
π
ν
ααα
λ
λ
ν
χ
αβ
βα
πσ
σµ
α
λ
ν
ν
ν
α
βα
σ
µ

76. DEFINITIONS (converted).
( )
( ) ( )( ) ( ) ( ) ( )
( )
( ) ( ) ( )( )
( )
( )
( ) ( ) ( ) ( ) ( ) ( )( )
.0;0;0
,;
.0;,
;
.0
.0),;(
.,),;(
21
2
ln
2
ln
2
lnlnln2lnln5.0
21
2
1
ln1ln1
2
ln2ln5.0
2
ln1
2
2ln
22
ln2
)ln()1()ln())(ln(
)(
)2ln(
2
1
2121
122112211
ln2
2
2
2
>>>∀
=
>∞∞−∈∀
=
≥∀=
≥∀=Γ
∞∞−∈∀=











 +
Γ+











Γ−











Γ−++−−++











 +
Γ+





+++











Γ+−
−−+−Γ−
−−+−Γ−







 −
+−
−
vvu
eUfdist
vt
eTtdist
xeX
xeX
xeXN
vvvv
uvvvvuvvvvv
v
etv
v
v
xx
xx
x
v
νν
ν
χ
βα
σµ
π
ννν
ν
βαβαα
σ
µ
πσ

77. PROPERTIES OF PROBABILITY DENSITY FUNCTIONS
1)(lim
0)(lim
)()(
0)(
=
=
=
ℜ∈∀≥
∞→
∞→
∞−
∫
xP
xp
dpxP
xxp
x
x
x
ττ

78. p(X) E(X) Var(X)
( )
( )
( )
( )
( )( )2
221
21
2
2
2
2
21
2
2
2
22
24
22
2
,;
2
0;
11;
);(
2)(
),;(
),;(
−−
−+
−
−
Γ
ννν
ννν
ν
ν
νν
ν
ν
ν
ααα
λλλ
ννχ
αβαββα
σµσµ
ν
Ffdist
Ttdist
Xedist
Xpoisson
X
X
XN
* From Schaum’s Outline of Probability and Statistics, second edition.

79. 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

80. DEFINITIONS
∫
∞
∞−
= dxxxpXE )()(
( ) ∫
∞
∞−
== dxxpeeEtM xtXt
X )()(
VARIANCE
)(:
]))([()]([)()(
2
222
XVarnote
XEXEXEXEXVar
=
−=−=
σ

81. STATISTICAL EXPECTATION ),;( 2
σµXN
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

82. 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

83. ),00.25;( 2
σµ =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

84. )(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

85. );( 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

86. 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. "

87. ∫ = dxXN ),00.25;( 2
σµ
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

88. ),00.25,( 2
σµ =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

89. THIS PAGE INTENTIONALLY LEFT BLANK

90. 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
( )X2
νχ

91. 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
( )∫ dxX2
νχ
ν=10.00

92. AN EXPRESSION OF THE CENTRAL LIMIT THEOREM
FROM HUNTSBERGER/BILLINGSLEY…
“ If n is large, then
n
X
σ
µ−
has approximately a standard normal
distribution, or (what is the same), X
has approximately a normal distribution
with mean μ and standard deviation
n
σ
. “

93. A percentage increase/decrease can be calculated as follows…
100×
−
OLD
OLDNEW
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…
1
5
5
5
510
==
−
Multiply by 100 and we get 1*100=100%...

94. θr 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
θr 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
θr 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
θr 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 rθ

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111. CALCBOOK

112. APPLICATION INPUT
SPECIFICATIONS
DESCRIPTION WORKING DIRECTORY
BSSLCALC BSSLMENU GENERATES SOLUTIONS TO BESSEL EQUATIONS C:SMITTYWORKBOOK
CALC2000 MENU2000 GENERATES STATISTICAL TABLES AND DIFFERENCE TABLES C:SMITTYWORKBOOK
CALC2010 MENU2010 GENERATES SOLTUIONS TO ORDINARY DIFFERENTIAL
EQUATIONS
C:SMITTYWORKBOOK
CALC2011 MENU2011 GENERATES STATISTICAL TABLES C:SMITTYWORKBOOK
CALC2012 MENU2012 TABULATES LOGARITHMIC FUNCTIONS C:SMITTYWORKBOOK
*The statistical formulae used in these algorithms comes from “Schaum’s Outline of Probability and Statistics”.

113. 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 OFTHE LORD IS, THERE IS OUR LIBERTY…”
(THUNDERING, & LIGHTNING, & VOICES) (16:18)

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

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

117. 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.

118. 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.

119. 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.

121. PSALMS 118:24
This is the day which the
LORD hath made; we will rejoice
and be glad in it.

122. 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.”

123. 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.”

125. PREPARED AND
EDITED
BY
KEITH SMITH
2012