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# how to display pdf file in picturebox in c# : Copy paste pdf text SDK application service wpf html web page dnn Financial%20Modeling%20with%20Excel%20and%20VBA6-part496

61
IV.  DISCRETE DISTRIBUTIONS
A.
Bernoulli Trials
Parameter p.
Probability density:
=
=
=
=
1
0
1
)
(
forx
p
forx
p
x
p X
Cumulative distribution function F(x):
p
F x
( )=1−
( )
Expected value of x:
p
E x
( )=
(
Variance of x:
)
(1
( )
p
p
V x
=
To generate a random number from a Bernoulli distribution:
EXCEL:
= IF( RAND() < p, 1, 0 )
VBA:
Function Random_Bernoulli( p As Double ) As Double
Dim u as Double
u = Rnd()
If u <= p Then
Random_Bernoulli = 0
Else
Random_Bernoulli = 1
End It
End Function
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62
B.
Binomial Distribution
Parameters p and n.
Number of successes in n independent trials, each with probability of success p.
Probability:
n x
x
p
p
x
n
x
P X
=
=
)
(1
)
(
Where:
)!
!(
!
x n n x
n
x
n
=
Cumulative distribution function F(x):
⎣ ⎦
n i
i
x
i
p
p
i
n
F x
=
=
)
(1
( )
0
Expected value of x:
np
E x
( )=
(
Variance of x:
)
(1
( )
p
np
V x
=
To generate a random number from a Binomial distribution?
EXCEL and VBA:
Series of Bernoulli Trials
Normal Approximation of the Binomial:
If n is large, then an approximation to B(n, p) is the normal distribution N( µ = np, σ =
np(1-p) ).  The approximation gets better as n increases.  So, to decide if n is large
enough to use the normal, both np and n(1 − p) must be greater than 5.
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63
C.
Trinomial Distribution
Parameters p, q and n.
The obvious way to extend the Bernoulli trials concept is to add ties.  Thus, each de
Moivre trial yields success ( x = 1 ), failure ( x = -1 ), or a tie ( x = 0 ).  The probability of
success is p, of failure q, and of a tie is 1 – p – q.
Probability:
n x x y
x y
p q
p q
x y y n n x x y
n
y
xY
P X
− −
− −
− −
=
=
=
)
(1
)!
! !(
!
)
,
(
Expected value of x and y:
nq
y
np
E x
=
Ε
=
( )
( )
Variance of x:
)
(1
( )
)
(1
( )
q
nq
p V V y
np
V x
=
=
To generate a random number from a de Moivre trial:
+ ≤
≤ +
=
1
ifp q q u
1
p q
ifp u
0
p
ifu
1
s
s
s
x
EXCEL and VBA:
Series of de Moivre Trials
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64
D.
Poisson Distribution
Parameter λ.
Number of events that occur in an interval of time when events are occurring at a
constant rate, say exponentially.
Probability:
{0,1,...}
!
( )
=
forx
x
e
Px
x
λλ
Cumulative Distribution Function:
⎣ ⎦
0
!
( )
0
=
=
if x
i
e
F x
x
i
i
λ
λ
Expected value of x and y:
λ
E(x)=
E
Variance of x:
λ
V(x)=
V
To generate a random number from a Poisson distribution:
Step 1
Let a = e
, b = 1, and i = 0.
Step 2
Generate u
s
and let b = b·u
s
.  If b < a, then return x = 1.
Otherwise, continue to Step 3.
Step 3
Set i = i + 1.  Go back to Step 2.
VBA:
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65
E.
Empirical Distributions
A cumulative distribution function gives the probability than a random variable X is less
that a given value x.  So,
)
(
( )
x
P X
F X
=
An empirical distribution uses actual data rather than a theoretical distribution function.
In the table below, n = 1000 observations are made with i = 4 outcomes, x
i
= { 100, 200,
1000, 5000 }.  The outcomes are sorted from low to high.  Then calculated is the
probability of each outcome, where the empirical probability, P(x
i
), is the ratio of the
count of each outcome, n
x
, to the total number of trials.
x
n
P(x
i
F(x
i
If u
s
= .71,
100
500
.50
.50
200
100
.10
.60
1000
250
.25
.85
x = 1000
5000
150
.15
1.0
Σ = 1000
Σ = 1.0
VBA:
Function Empirical() As Double
Dim us As Double
us = Rnd()
Dim x As Double
Select Case us
Case Is < 0.5
x = 100
Case 0.5 To 0.6
x = 200
Case 0.6 To 0.85
x = 1000
Case 0.85 To 1
x = 5000
End Select
Empirical = x
End Function
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66
F.
Linear Interpolation
Linear interpolation is a method of curve fitting using linear polynomials.  If the coordinates of
two points are known, the linear interpolant is the straight line between these points. For a point
in the interval, the coordinate values, x and y, along the straight line are given by:
0
1
0
0
1
0
x
x
x x
y
y
y y
=
Solving for y we get:
)
(
0
0
1
0
1
0
x x
x
x
y
y
y y
+
=