Mathematicalmethods Lecture Slides Week4 MultipleIntegrals
Transcript of Mathematicalmethods Lecture Slides Week4 MultipleIntegrals
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AMATH 460: Mathematical Methods
for Quantitative Finance
4. Multiple Integrals
Kjell KonisActing Assistant Professor, Applied Mathematics
University of Washington
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Outline
1 Double Integrals
2 Fubiniβs Theorem
3 Change of Variables for Double Integrals
4 Change of Variables Example
5 Double Integrals of Separable Functions
6 Polar Coordinates
7 A Culturally Important Integral
8 Marginal Density of a Bivariate Normal Distribution
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Outline
1 Double Integrals
2 Fubiniβs Theorem
3 Change of Variables for Double Integrals
4 Change of Variables Example
5 Double Integrals of Separable Functions
6 Polar Coordinates
7 A Culturally Important Integral
8 Marginal Density of a Bivariate Normal Distribution
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Double Integrals
Review of the definite integral of a single-variable function
a b
f(x)
b a
f (x ) dx = limnββ
nβ1i =0
f (a + i βx ) βx βx = b β a
n
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Double Integrals
In general, double integral over a rectangle R = [a , b ] Γ [c , d ] R
f (x , y ) dA = limm,nββ
mi =1
n j =1
f (a + i βx , c + j βy ) βA
If f (x , y ) β₯ 0 β(x , y ) β R , then
V =
R f (x , y ) dA
is the volume of the region above R and below surface z = f (x , y )
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Properties of Double Integrals
Linearity Properties: R
[f (x , y ) + g (x , y )] dA =
R
f (x , y ) dA +
R
g (x , y ) dA
R
cf (x , y ) dA = c R
f (x , y ) dA
Comparison:
If f (x , y )
β₯g (x , y )
β(x , y )
βR then
R f (x , y ) dA β₯
R
g (x , y ) dA
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Iterated Integrals
Suppose f (x , y ) is continuous on the rectangle R = [a , b ] Γ [c , d ]Partial integration: fix x , integrate f (x , y ) as a function of y alone
A(x ) =
d c
f (x , y ) dy
An iterated integral is the integral of A(x ) wrt x
b a
A(x ) dx = b a
d c
f (x , y ) dy
dx
Usually the brackets are omitted
b a d c
f (x , y ) dy dx = b a d
c f (x , y ) dy dx
Iterating the other way
d
c b
af (x , y ) dx dy =
d
c
b
af (x , y ) dx
dy
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Double Integrals vs. Iterated Integrals
Big Question:
What is the relationship between a double integral and an iterated
integral? R
f (x , y ) dA ?
b a
d c
f (x , y ) dy dx
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Outline
1 Double Integrals
2 Fubiniβs Theorem
3 Change of Variables for Double Integrals
4 Change of Variables Example
5 Double Integrals of Separable Functions
6 Polar Coordinates
7 A Culturally Important Integral
8 Marginal Density of a Bivariate Normal Distribution
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Fubiniβs Theorem
If f
(x , y
) is continuous on the rectangle R
= [a , b
] Γ [c , d
] then R
f (x , y ) dA =
b a
d c
f (x , y ) dy dx =
d c
b a
f (x , y ) dx dy
The order of iteration does not matter
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Example
Let R = [1, 3]Γ [2, 5] and f (x , y ) = 2y β 3x . Compute
R f (x , y ) dA
R
f (x , y ) dA =
5
2
3
1(2y β 3x ) dx dy
=
52
2xy β 3
2x 2
3
1
dy
=
52
6y β 27
2
β
2y β 32
dy
= 5
24y β 12 dy
=
2y 2 β 12y
5
2
=50 β 60β 8 β 24 = 6
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Example (continued)
R
f (x , y ) dA = 3
1
5
2
(2y
β3x ) dy dx
=
31
(y 2 β 3xy )
5
2
dx
= 3
1[(25 β 15x ) β (4 β 6x )] dx
=
31
21β 9x dx
=
21x β 9
2x 2 3
1
= 63 β 81
2 β 21 β 9
2 = 42 β 36 = 6Kjell Konis (Copyright Β© 2013) 4. Multiple Integrals 13 / 58
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Double Integrals Non-Rectangular Regions
If f (x , y ) is continuous on a region D that can be described
D = {(x , y ) : a β€ x β€ b , g 1(x ) β€ y β€ g 2(x ) }
then
D
f (x , y ) dA = b
a g 2(x )
g 1(x )
f (x , y ) dy dx
If f (x , y ) is continuous on a region D that can be described
D = {(x , y ) : c β€ y β€ d , h1(y ) β€ x β€ h2(y ) }then
D f (x , y ) dA =
d c
h2(x )h1(x )
f (x , y ) dx dy
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Example
Let D ={
(x , y ) :|x
|+|y
| β€1}diamond w/ corners at (0,Β±1) and (Β±1, 0)
Compute the integral of f (x , y ) = 1 over D
D
1 dA
= 0
β1
1+x
β1βx
1 dy dx + 1
0 1βx
x
β1
1 dy dx
=
0β1
y
1+x
β1βx
dx +
10
y
1βx
x β1
dx
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( )
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Example (continued)
= 0β1
1 + x
β β 1 β x dx + 1
0
1 β x β x β 1 dx =
0β1
2 + 2x
dx +
10
2 β 2x dx
=2x + x 2
0β1
+2x β x 21
0
= 0 + 0β β 2 + 1 + 2 β 1β 0 β 0= 1 + 1
= 2
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Outline
1 Double Integrals
2 Fubiniβs Theorem
3 Change of Variables for Double Integrals
4 Change of Variables Example
5 Double Integrals of Separable Functions
6 Polar Coordinates
7 A Culturally Important Integral
8 Marginal Density of a Bivariate Normal Distribution
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Ch f V i bl Si l V i bl C
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Change of Variables: Single Variable Case
Let f (x ) be a continuous function
Let g (s ) be a continuously differentiable and invertible functionβ’ Implies g (s ) either strictly increasing or strictly decreasing
g (s ) maps the interval [c , d ] into the interval [a , b ], i.e.,
s β [c , d ] β x = g (s ) β [a , b ]
Integration by substitution says:
x =b
x =af (x ) dx =
s =g
β1
(b )
s =g β1(a)f (g (s )) g (s ) ds
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Ch f V i bl F i f 2 V i bl
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Change of Variables: Functions of 2 Variables
Let f (x , y ) be a continuous function
Want to compute:
D f (x , y ) dA
Let β¦ be a domain such that the mapping
x = x (s , t ) y = y (s , t )
of a point (s , t ) β β¦ to a point (x , y ) β D is one-to-one and ontoβ’ x (s , t ) and y (s , t ) continuously differentiable
That is,
(s , t ) β β¦ ββ (x , y ) = (x (s , t ), y (s , t )) β D Want to find a function h(s , t ) such that
D f (x , y ) dx dy =
β¦h(s , t ) ds dt
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Change of Variables
Get startedf (x , y ) = f (x (s , t ), y (s , t ))
In the single variable case, if x = g (s ) then
dx = g (s ) ds
In the 2-variable case, (x , y ) = (x (s , t ), y (s , t )) is a vector-valuedfunction of 2 variables
The gradient of (x (s , t ), y (s , t )) is the 2 Γ 2 array
D (x (s , t ), y (s , t )) =
β x β s
β x β t
β y
β s
β y
β t
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J bi
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Jacobian
The 2-variable equivalent of dx = g (s ) ds is
dx dy =
β x
β s
β y
β t β β x
β t
β y
β s
ds dt
and the quantity in the square brackets is called the Jacobian
2 dimensional change of variables formula:
D
f (x , y ) dx dy =
β¦f (x (s , t ), y (s , t ))
β x β s β y β t β β x β t β y β s ds dt
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Example
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Example
Example from previous sectionLet D = {(x , y ) : |x | + |y | β€ 1}diamond w/ corners at (0,Β±1) and (Β±1, 0)Compute the integral of f (x , y ) = 1 over D
D
1 dA = 0β1
1+x β1βx 1 dy
dx +
10
1βx x β1 1 dy
dx
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Example (continued)
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Example (continued)
Consider the change of variables:
s = x + y , t = x β y
Solve for x and y in terms of s and t :
x = s + t
2 , y =
s β t 2
Compute the partial derivatives of the change of variables:
β x
β s =
1
2,
β x
β t =
1
2,
β y
β s =
1
2,
β y
β t = β1
2
Compute the Jacobian:
β x
β s
β y
β t β β x
β t
β y
β s =
1
2
β1
2
β 1
2
1
2 = β1
4 β 1
4 = β1
2
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Example (continued)
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Example (continued)
Multivariate change of variables formula:
D
1 dx dy =
β¦1β x β s β y β t β β x β t β y β s ds dt = β¦ 12 ds dt
Finally, domain of integration (β¦):
s = x + y
t = x β y
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Example (continued)
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Example (continued)
Domain of integration: β¦ = [β1, 1] Γ [β1, 1]
D 1 dA =
0
β1
1+
x
β1βx 1 dy
dx +
1
0
1β
x
x β11 dy
dx
=
β¦
1
2 ds dt
=
t =1t =β1
s =1s =β1
1
2 ds
dt
= t =1
t =β1 s
2s =1
s =β1 dt
=
t =1t =β1
1 dt
= 2
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Outline
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Outline
1 Double Integrals
2 Fubiniβs Theorem
3 Change of Variables for Double Integrals
4 Change of Variables Example
5 Double Integrals of Separable Functions
6 Polar Coordinates
7 A Culturally Important Integral
8 Marginal Density of a Bivariate Normal Distribution
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Change of Variables Example
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Change of Variables Example
Evaluate
D
x dx dy where
D =
(x , y ) β R2 : x β₯ 0,
1 β€ xy β€ 2,
1 β€ y x β€ 2
Can assume x > 0
D = 1
x β€y β€
2
x x β€ y β€ 2x x
y
y =1
x
y =2
x
y = x
y = 2x
D x dx dy = 1
β 2
2 2x
1x
x dy dx + β
2
1 2x
x x dy dx
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x
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=
β
22
1x
x dy
dx +
β1
x
x x dy
dx
= 1β
22
xy
y =2x
y = 1x
dx +
β 21
xy
y = 2
x
y =x
dx
=
1β
22
2x 2 β 1
dx +
β 21
2 β x 2
dx
=2
3x 3 β x
1
β 2
2
+2x β 1
3x 3β
2
1
=
23 β 1β 23 (β 2)
3
23 β β 22
+
2β 2 β 13 (β 2)3β 2 β 13
= β13 β
β 2
6 +
3β
2
6 +
12β
2
6 β 4
β 2
6 β 5
3 = β12 + 10β 2
6 =
β6 + 5β 23
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Again Using a Change of Variables
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Again, Using a Change of Variables
Consider the change of variables
s = xy , t = y x
D =
(x , y ) β R2 : x β₯ 0,
1 β€ xy β€ 2,
1 β€ y x β€ 2
β¦ = [1, 2] Γ [1, 2] x
y
y =1
x
y =2
x
y = x
y =
2x
D
x dx dy =
t =2t =1
s =2s =1
x dx dy
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Again, Using a Change of Variables
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Again, Using a Change of Variables
First, solve for functions x = x (s , t ) and y = y (s , t ):
x = s
t , y = β st
Partial derivatives for the change of variables are:
β x
β s =
β
β s
s
12
t β12
= 1
2s β
12 t β
12 =
1
2β
st
β y
β s = β
β s
s 1
2 t 1
2
= 1
2s β
12 t
12 =
β t
2β
s
β x
β t =
β
β t
s
12
t β12
= β12
s 12 t β
32 = β
β s
2t β
t
β y β t = β β t
s
12 t
12
= 1
2s
12 t β
12 =
β s
2β
t
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Again, Using a Change of Variables
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Again, Using a Change of Variables
Jacobian for the change of variables:
β x
β s
β y
β t β β x
β t
β y
β s =
1
2β
st
β s
2β
t ββ
β s
2t β
t
β t
2β
s
=
1
4t +
1
4t
= 1
2t
Change of variables formula: D
x dx dy =
t =2t =1
s =2s =1
s
t
1
2t ds dt
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ds
dt
1 t =2 s =2
s12 tβ
32 ds
dt
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t =1
s =1
t 2t
ds
dt =
2
t =1
s =1
s 2 t 2 ds
dt
= 1
2 t =2
t =1 2
3s
32 t β
32
s =2
s =1 dt
= 1
3
t =2t =1
2β
2t β32 β t β 32
dt
= 13 t =2t =1
2β 2 β 1t β 32 dt
= 2β
2 β 13
β2t β 12
t =2
t =1
= 2β 2 β 1
3 (2 β
β 2)
= 5β
2 β 63Kjell Konis (Copyright Β© 2013) 4. Multiple Integrals 32 / 58
Outline
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1 Double Integrals
2 Fubiniβs Theorem
3 Change of Variables for Double Integrals
4 Change of Variables Example
5 Double Integrals of Separable Functions
6 Polar Coordinates
7 A Culturally Important Integral
8 Marginal Density of a Bivariate Normal Distribution
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Separable Functions
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p
Take another look at the example in the last section
t =2t =1
s =2s =1
s t
12t
ds
dt = 12
t =2t =1
s =2s =1
s 12 t β 32 ds
dt
Since t (and any function of t ) is constant while integrating wrt s t =2
t =1
s =2
s =1
s
t
1
2t ds
dt =
1
2
t =2
t =1t β
32
s =2
s =1s
12 ds
dt
The double integral is the product of 2 single-variable definiteintegrals
t =2t =1
s =2s =1
s
t
1
2t ds
dt =
1
2
s =2s =1
s 12 ds
t =2t =1
t β32 dt
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Separable Functions
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p
1
2 s =2
s =1s
12 ds
t =2
t =1t β
32 dt =
1
2 2
3s
32
s =2
s =1 β2t
β 12
t =2
t =1
= β2
3
s
32
s =2
s =1
t β
12
t =2
t =1
= β23
2β 2 β 1 1β 2 β 1
= β23
2β 2
β 2 β 1β
2+ 1
= β13
4β 4β 2 ββ 2 + 2
= 5β
2 β 63
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Separable Functions
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Let R = [a , b ] Γ [c , d ] be a rectangleLet f (x , y ) be a continuous, separable function
f (x , y ) = g (x ) h(y )
g (x ) and h(x ) continuous
Double integral of f over R is the product of single-variable integrals R
f (x , y ) dx dy =
d c
b a
g (x ) h(y ) dx dy
= d c
h(y ) b
ag (x ) dx
dy
=
b
ag (x ) dx
d
c h(y ) dy
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Outline
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1 Double Integrals
2 Fubiniβs Theorem
3 Change of Variables for Double Integrals
4 Change of Variables Example
5 Double Integrals of Separable Functions
6 Polar Coordinates
7 A Culturally Important Integral
8 Marginal Density of a Bivariate Normal Distribution
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Polar Coordinates
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Describe points in R2 using
r β [0, β)ΞΈ β [0, 2Ο)r = distance from origin
ΞΈ = angle counter clockwisefrom positive x -axis
(x , y ) ββ (r , ΞΈ)x (r , ΞΈ) = r cos(ΞΈ)
y (r , ΞΈ) = r sin(ΞΈ)
Can simplify integrationproblems
x
y
( , )x0 y0
r0
ΞΈ0
( , )x1 y1
r1
ΞΈ1
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Change of Variables to Polar Coordinates
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The change of variables is:
x (r , ΞΈ) = r cos(ΞΈ) y (r , ΞΈ) = r sin(ΞΈ)
The partial derivatives of the change of variables are:
β x
β r
= cos(ΞΈ) β x
βΞΈ
=
βr sin(ΞΈ)
β y
β r
= sin(ΞΈ) β y
βΞΈ
= r cos(ΞΈ)
The Jacobian is:
β x β r
β y βΞΈ β β x
βΞΈβ y β r
= cos(ΞΈ) Β· r cos(ΞΈ) β (βr sin(ΞΈ)) Β· sin(ΞΈ)
= r
cos2(ΞΈ) + sin2(ΞΈ)
= r
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Change of Variables to Polar Coordinates
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The change of variable formula to polar coordinates:
D
f (x , y ) dx dy =
D f (r cos(ΞΈ), r sin(ΞΈ)) r dr d ΞΈ
Integrate f (x , y ) over a disk of radius R centered at the origin:
D (0,R )
f (x , y ) dx dy =
2Ο
0
R
0f (r cos(ΞΈ), r sin(ΞΈ)) r dr
d ΞΈ
Integrate f (x , y ) over the plane R2:
R2
f (x , y ) dx dy =
2Ο0
β0
f (r cos(ΞΈ), r sin(ΞΈ)) r dr
d ΞΈ
The latter can be useful for integrating probability density functions
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Example
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Let D = {(x , y ) : x 2 + y 2 β€ 1} (disk of radius 1 centered at origin)
Compute the integral D
(1 β x 2 β y 2) dx dy
Try once using xy-coordinates
Then try once using polar coordinates
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Df (x , y ) dy dx =
1
β1
β 1βx 2ββ 1βx
2(1 β x 2 β y 2) dy
dx
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D
1
β
1 x
= 1
β1 (1 βx 2)y
β y 3
3
y =β
1βx 2
y =ββ 1βx 2 dx
=
1β1
2(1 β x 2)
1 β x 2 β 2(
β 1 β x 2)3
3
dx
=
1
β1
6(β 1 β x 2)3 β 2(β 1 β x 2)3
3
dx
= 4
3 1
β1 ( 1 β
x 2)3 dx ...
...
= 1
6x 1 β
x 2(5
β2x 2) + 3 arcsin(x )
1
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Example (in polar coordinates)
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D f (x , y ) dy dx =
2Ο0
101 β r
2
cos2
(ΞΈ) β r 2
sin2
(ΞΈ)
r dr d ΞΈ
=
2Ο0
10
1 β r 2 cos2(ΞΈ) + sin2(ΞΈ) r dr d ΞΈ
= 2Ο
0
10
r β r 3 dr d ΞΈ
=
2Ο0
r 2
2 β r
4
4
1
0
d ΞΈ
=
2Ο0
1
4 d ΞΈ =
Ο
2
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Outline
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1 Double Integrals
2 Fubiniβs Theorem
3 Change of Variables for Double Integrals
4 Change of Variables Example
5 Double Integrals of Separable Functions
6 Polar Coordinates
7 A Culturally Important Integral
8 Marginal Density of a Bivariate Normal Distribution
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The Standard Normal Density
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The standard normal density
Ο(x ) = 1β
2Οe β
x 2
2
The function Ξ¦ used in the Black-Scholes formula
Ξ¦(x ) = x
ββΟ(t ) dt
Raises 2 12 questions:
1 Where does the 1β 2Ο come from?
2 Why not use a closed-form expression for Ξ¦?
2 12 Where does the 1β
2Ο come from?
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The Standard Normal Density
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Since Ο is a probability density function
βββ
Ο(x ) dx = βββ
Ο(x ) dx = 1
Implies that βββ
e β x 22 dx = β 2Ο
Change of variables
βββ e β
x 2
dx = β Ο
The problem is e βx 2 does not have an antiderivative
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Change to Polar Coordinates
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Let
M = β
ββe βx
2dx
Want to show that M =β Ο
Can also express M as
M =
βββ
e βy 2
dy
Now want to show that M 2
= Ο
M 2 =
βββ
e βx 2
dx
βββ
e βy 2
dy
This is the double integral of a separable function
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Change to Polar Coordinates
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Unseparate the double integral
M 2 = β
ββe βx
2dx
β
ββe βy
2dy
=
βββ
e βx 2 βββ
e βy 2
dy
dx
= βββ e βx 2 βββ e βy
2dy
=
βββ
βββ
e β(x 2+y 2) dy dx
=
R2e β
(x 2+y 2)dy dx
Change to polar coordinates
=
2Ο
0 β
0e β[r cos(ΞΈ)]
2+[r sin(ΞΈ)]2 r dr d ΞΈ
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Change to Polar Coordinates
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M 2 = 2Ο
0 β
0e β[r cos(ΞΈ)]
2+[r sin(ΞΈ)]2 r dr d ΞΈ
=
2Ο0
β0
e βr 2[cos2(ΞΈ)+sin2(ΞΈ)]r dr d ΞΈ
= 2Ο
0 β
0
e βr 2
r dr d ΞΈ let u =β
r 2
=
2Ο0
d ΞΈ
β1
2
u =ββu =0
e u (β2r dr )
du = β2r dr
= ΞΈ
2Ο
0
β12 e u ββ
0
= [2Ο β 0]β1
2
limt βββ e
t β 1
= 2Ο Β· 12
= Ο
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Change to Polar Coordinates
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In summary:
Started out with
M =
βββ
e βx 2
dx
Showed that M 2 = Ο and thus that M =β Ο
Can conclude that
βββ e βx 2
dx = β Ο
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Outline
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1 Double Integrals
2 Fubiniβs Theorem
3 Change of Variables for Double Integrals
4
Change of Variables Example5 Double Integrals of Separable Functions
6 Polar Coordinates
7 A Culturally Important Integral
8 Marginal Density of a Bivariate Normal Distribution
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Marginal Density of a Bivariate Normal Distribution
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Bivariate normal density function: f X ,Y (x , y ;Β΅x , Β΅y , Οx , Οy , Ο)
1
2ΟΟx Οy
1β Ο2 exp
β
(x β Β΅x )2Ο2x
β 2Ο(x β Β΅x )(y β Β΅y )Οx Οy
+ (y β Β΅y )2
Ο2y
2(1 β Ο2)
The marginal density is
f Y (y ) = β
ββf X ,Y (x , y ) dx
Let X and Y be returns on an asset in consecutive periods
Assume that Β΅x = Β΅y = Β΅ and Οx = Οy = Ο
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Marginal Density β
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f X (x ) =
βββ
f X ,Y (x , y ) dy
= βββ
12ΟΟ2
1 β Ο2 exp
β(x β Β΅)2 β 2Ο(x β Β΅)(y β Β΅) + (y β Β΅)2
2Ο2(1 β Ο2)
dy
Guess that f X (x ) is normal with mean Β΅ and variance Ο2
f X (x ) = 1β
2ΟΟexp
β(x β Β΅)
2
2Ο2
Γ βββ
C exp (x β Β΅)22Ο2
β (x β Β΅)2
β 2Ο(x β Β΅)(y β Β΅) + (y β Β΅)2
2Ο2(1 β Ο2)
dy
where C = 1β 2ΟΟ 1 β
Ο2
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Marginal Density
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Lets look at the quantity in the square brackets
(x β Β΅)2
2Ο2 β (x
βΒ΅)2
β2Ο(x
βΒ΅)(y
βΒ΅) + (y
βΒ΅)2
2Ο2(1 β Ο2)
=
(1 β Ο2)(x β Β΅)2 β (x β Β΅)2 + 2Ο(x β Β΅)(y β Β΅) β (y β Β΅)2
2Ο2(1 β Ο2)
=
βΟ2(x β Β΅)2 + 2Ο(x β Β΅)(y β Β΅) β (y β Β΅)2
2Ο2(1 β Ο2)
=β
Ο2(x
βΒ΅)2
β2Ο(x
βΒ΅)(y
βΒ΅) + (y
βΒ΅)2
2Ο2(1 β Ο2)
=
βΟ(x β Β΅) β (y β Β΅)2
2Ο2(1 β Ο2)
=
β
y β (Β΅ + Ο(x β Β΅))22(Ο 1 β Ο
2)2
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Marginal Density
2
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f X (x ) = 1β
2ΟΟexp
β(x β Β΅)
2
2Ο2
Γ βββ
C exp
(x β Β΅)22Ο2
β (x β Β΅)2 β 2Ο(x β Β΅)(y β Β΅) + (y β Β΅)2
2Ο2(1 β Ο2)
dy
Lets look just at the integrand
1β 2Ο(Ο
1 β Ο2) exp
β
y β (Β΅ + Ο(x β Β΅))22(Ο
1 β Ο2)2
Let m = (Β΅ + Ο(x β Β΅)) and s = (Ο 1 β Ο2), the integrand becomes1β 2Οs
exp
β(y βm)
2
2s 2
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Marginal Density
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The integral becomes
βββ
1β 2Οs
expβ(y βm)2
2s 2
dy
Integral of a normal density over the real line, thus equal to 1
The guess for the marginal density was correct
f X (x ) = 1β
2ΟΟexp
β(x β Β΅)
2
2Ο2
β
ββ
1β 2Οs
exp
β(y βm)
2
2s 2
dy
= 1β
2ΟΟexp
β(x β Β΅)
2
2Ο2
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Bonus: Conditional Density Function
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The function that we integrated out is the conditional density
Denoted by f Y |X (y |x )
f Y |X (y |x ) = 1β
2Ο(Ο
1 β Ο2) expβ
y β (Β΅ + Ο(x β Β΅))22(Ο
1β Ο2)2
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