probability theory presentation 09
TRANSCRIPT
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BST 401 Probability Theory
Xing Qiu Ha Youn Lee
Department of Biostatistics and Computational BiologyUniversity of Rochester
October 5, 2010
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Outline
1 Product Measures and Fubinis Theorem
2 Kolmogorovs Extension Theorem
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Product measure space
The usual Euclidean space equipped with the usual
Lebesgue measure, (Rn,B(Rn, (n)) is a product ofseveral (R,B, ).
Definition:the wholespace: Cartesian product.the -algebra: minimum -algebra that contains allmeasurable rectangles, which are sets that have formA1 A2 . . . An, A1, . . . , An are measurable sets in eachone-dim space.the product measure: well defined on measurablerectangles, (A1 A2 . . . An) = 1(A1)2(A2) . . . n(An).
Qiu, Lee BST 401
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Product measure space
The usual Euclidean space equipped with the usual
Lebesgue measure, (Rn,B(Rn, (n)) is a product ofseveral (R,B, ).
Definition:the wholespace: Cartesian product.the -algebra: minimum -algebra that contains allmeasurable rectangles, which are sets that have formA1 A2 . . . An, A1, . . . , An are measurable sets in eachone-dim space.the product measure: well defined on measurablerectangles, (A1 A2 . . . An) = 1(A1)2(A2) . . . n(An).
Qiu, Lee BST 401
http://goforward/http://find/http://goback/ -
8/8/2019 Probability Theory Presentation 09
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Product measure space
The usual Euclidean space equipped with the usual
Lebesgue measure, (Rn,B(Rn, (n)) is a product ofseveral (R,B, ).
Definition:the wholespace: Cartesian product.the -algebra: minimum -algebra that contains allmeasurable rectangles, which are sets that have formA1 A2 . . . An, A1, . . . , An are measurable sets in eachone-dim space.the product measure: well defined on measurablerectangles, (A1 A2 . . . An) = 1(A1)2(A2) . . . n(An).
Qiu, Lee BST 401
http://goforward/http://find/http://goback/ -
8/8/2019 Probability Theory Presentation 09
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Product measure space
The usual Euclidean space equipped with the usual
Lebesgue measure, (Rn,B(Rn, (n)) is a product ofseveral (R,B, ).
Definition:the wholespace: Cartesian product.the -algebra: minimum -algebra that contains allmeasurable rectangles, which are sets that have formA1 A2 . . . An, A1, . . . , An are measurable sets in eachone-dim space.the product measure: well defined on measurablerectangles, (A1 A2 . . . An) = 1(A1)2(A2) . . . n(An).
Qiu, Lee BST 401
http://goforward/http://find/http://goback/ -
8/8/2019 Probability Theory Presentation 09
7/45
Product measure space
The usual Euclidean space equipped with the usual
Lebesgue measure, (Rn,B(Rn, (n)) is a product ofseveral (R,B, ).
Definition:the wholespace: Cartesian product.the -algebra: minimum -algebra that contains allmeasurable rectangles, which are sets that have formA1 A2 . . . An, A1, . . . , An are measurable sets in eachone-dim space.the product measure: well defined on measurablerectangles, (A1 A2 . . . An) = 1(A1)2(A2) . . . n(An).
Qiu, Lee BST 401
http://goforward/http://find/http://goback/ -
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The Classical Product Measure Theorem
Calculus analogy: double integral. Draw a diagram.
For E F1F2, let Ex (Ey) be the cross-section at x (y),
Ex = {y : (x, y) E} , Ey = {x : (x, y) E} .
The measure of E can be calculated by iterated integration:
(E) =
1
2(Ex)d1(x) =
2
1(Ey)d2(y).
There is one and only one measure which satisfies(A1 A2) = 1(A1)2(A2). (There is a mathematicalterminology for this property. () is called the tensorproduct of 1() and (), denoted by 1 2).
Qiu, Lee BST 401
http://goforward/http://find/http://goback/ -
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The Classical Product Measure Theorem
Calculus analogy: double integral. Draw a diagram.
For E F1F2, let Ex (Ey) be the cross-section at x (y),
Ex = {y : (x, y) E} , Ey = {x : (x, y) E} .
The measure of E can be calculated by iterated integration:
(E) =
1
2(Ex)d1(x) =
2
1(Ey)d2(y).
There is one and only one measure which satisfies(A1 A2) = 1(A1)2(A2). (There is a mathematicalterminology for this property. () is called the tensorproduct of 1() and (), denoted by 1 2).
Qiu, Lee BST 401
http://goforward/http://find/http://goback/ -
8/8/2019 Probability Theory Presentation 09
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The Classical Product Measure Theorem
Calculus analogy: double integral. Draw a diagram.
For E F1F2, let Ex (Ey) be the cross-section at x (y),
Ex = {y : (x, y) E} , Ey = {x : (x, y) E} .
The measure of E can be calculated by iterated integration:
(E) =
1
2(Ex)d1(x) =
2
1(Ey)d2(y).
There is one and only one measure which satisfies(A1 A2) = 1(A1)2(A2). (There is a mathematicalterminology for this property. () is called the tensorproduct of 1() and (), denoted by 1 2).
Qiu, Lee BST 401
http://goforward/http://find/http://goback/ -
8/8/2019 Probability Theory Presentation 09
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Fubinis Theorem
Motivation: f(x, y) defined on D = A B. How do we knowthat
D f(x, y)dxdy can be computed by iterated
integration:
A
B
f(x, y)dydx?
The answer: D |f(x, y)
|dxdy