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BST 401 Probability Theory
Xing Qiu Ha Youn Lee
Department of Biostatistics and Computational BiologyUniversity of Rochester
September 21, 2009
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Outline
1 Lebesgue-Stieltjes Measure and Distribution Functions
2 Measurable functions
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Lebesgue measure review, the construction
Let F0 be the field generated by the collection of all
intervals, 0 be the usual length measure of intervals.
Extend 0 to 1 : G R, where G is F0 plus limiting sets
of F0, 1 on these limiting sets are defined by exchangethe order of limit and measure.
Extend 1 to , which is an outer measure defined on 2.
Unfortunately, in general does not satisfy
countable-additivity.
Restrict to the collection of measurable sets, denoted by
F, which is a -algebra.
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Lebesgue measure review, the construction
Let F0 be the field generated by the collection of all
intervals, 0 be the usual length measure of intervals.
Extend 0 to 1 : G R, where G is F0 plus limiting sets
of F0, 1 on these limiting sets are defined by exchangethe order of limit and measure.
Extend 1 to , which is an outer measure defined on 2.
Unfortunately, in general does not satisfy
countable-additivity.
Restrict to the collection of measurable sets, denoted by
F, which is a -algebra.
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Lebesgue measure review, the construction
Let F0 be the field generated by the collection of all
intervals, 0 be the usual length measure of intervals.
Extend 0 to 1 : G R, where G is F0 plus limiting sets
of F0, 1 on these limiting sets are defined by exchangethe order of limit and measure.
Extend 1 to , which is an outer measure defined on 2.
Unfortunately, in general does not satisfy
countable-additivity.
Restrict to the collection of measurable sets, denoted by
F, which is a -algebra.
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Lebesgue measure review, the construction
Let F0 be the field generated by the collection of all
intervals, 0 be the usual length measure of intervals.
Extend 0 to 1 : G R, where G is F0 plus limiting sets
of F0, 1 on these limiting sets are defined by exchangethe order of limit and measure.
Extend 1 to , which is an outer measure defined on 2.
Unfortunately, in general does not satisfy
countable-additivity.
Restrict to the collection of measurable sets, denoted by
F, which is a -algebra.
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Lebesgue measure review, the main results
The Carathodory extension theorem. There exists one
and only one way to extend a -finite measure 0 on an
algebra F0 to, F,the -algebra generated by F0.
The measure approximation theorem. For any A F anda given > 0, there exists a set B F0 such that(AB) < .
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Lebesgue measure review, the main results
The Carathodory extension theorem. There exists one
and only one way to extend a -finite measure 0 on an
algebra F0 to, F,the -algebra generated by F0.
The measure approximation theorem. For any A F anda given > 0, there exists a set B F0 such that(AB) < .
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Generalizations
Slight generalization of Lebesgue Measure:
((a, b]) = F(b) F(a), where F() is a non-decreasing,continuous function.
Further generalization: F() just needs to be a
right-continuous function. So jumps are allowed.
Def. of right-continuity: F(xn) F(x) when xn x.
Such an F is called a Stieltjes measure function. If is a
probability measure, it is called the distribution function of
. We will use these two terms interchangeably.Theorem (1.5), pg. 440. Given such an F, there is a
measure s.t. ((a, b]) = F(b) F(a).
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Generalizations
Slight generalization of Lebesgue Measure:
((a, b]) = F(b) F(a), where F() is a non-decreasing,continuous function.
Further generalization: F() just needs to be a
right-continuous function. So jumps are allowed.
Def. of right-continuity: F(xn) F(x) when xn x.
Such an F is called a Stieltjes measure function. If is a
probability measure, it is called the distribution function of
. We will use these two terms interchangeably.Theorem (1.5), pg. 440. Given such an F, there is a
measure s.t. ((a, b]) = F(b) F(a).
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Generalizations
Slight generalization of Lebesgue Measure:
((a, b]) = F(b) F(a), where F() is a non-decreasing,continuous function.
Further generalization: F() just needs to be a
right-continuous function. So jumps are allowed.
Def. of right-continuity: F(xn) F(x) when xn x.
Such an F is called a Stieltjes measure function. If is a
probability measure, it is called the distribution function of
. We will use these two terms interchangeably.Theorem (1.5), pg. 440. Given such an F, there is a
measure s.t. ((a, b]) = F(b) F(a).
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Generalizations
Slight generalization of Lebesgue Measure:
((a, b]) = F(b) F(a), where F() is a non-decreasing,continuous function.
Further generalization: F() just needs to be a
right-continuous function. So jumps are allowed.
Def. of right-continuity: F(xn) F(x) when xn x.
Such an F is called a Stieltjes measure function. If is a
probability measure, it is called the distribution function of
. We will use these two terms interchangeably.Theorem (1.5), pg. 440. Given such an F, there is a
measure s.t. ((a, b]) = F(b) F(a).
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Generalizations
Slight generalization of Lebesgue Measure:
((a, b]) = F(b) F(a), where F() is a non-decreasing,continuous function.
Further generalization: F() just needs to be a
right-continuous function. So jumps are allowed.
Def. of right-continuity: F(xn) F(x) when xn x.
Such an F is called a Stieltjes measure function. If is a
probability measure, it is called the distribution function of
. We will use these two terms interchangeably.Theorem (1.5), pg. 440. Given such an F, there is a
measure s.t. ((a, b]) = F(b) F(a).
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Definitions
The converse of Theorem (1.5) is almosttrue. For most
measures we can define its distribution function. The only
exceptions: ((a, b]) = for some finite interval.
Definition
A Lebesgue-Stieltjes measure on R is a measure : B Rsuch that (I) < for each bounded interval.
Alternatively, we may define L-S measure by F() whichsatisfies the non-decreasing and right-continuity conditions.
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Comments and examples
Page 1.4.5.
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Discrete measure
Page 26.
Let be a L-S measure that is concentrated on a
countable set S = {x1, x2, . . . , }.
Distribution function: step function.
can be extended to 2.
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Discrete measure
Page 26.
Let be a L-S measure that is concentrated on a
countable set S = {x1, x2, . . . , }.
Distribution function: step function.
can be extended to 2.
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Discrete measure
Page 26.
Let be a L-S measure that is concentrated on a
countable set S = {x1, x2, . . . , }.
Distribution function: step function.
can be extended to 2.
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Restriction
In the discrete measure example, we can restrict to 2S
instead of B. It pretty much has the same property.
Remark: why we dont say restrict on S?Another restriction example: is concentrated on some
interval [a, b].
Construction of B[a, b]. Then we can restrict on this-field without loosing its mathematical properties.
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Restriction
In the discrete measure example, we can restrict to 2S
instead of B. It pretty much has the same property.
Remark: why we dont say restrict on S?Another restriction example: is concentrated on some
interval [a, b].
Construction of B[a, b]. Then we can restrict on this-field without loosing its mathematical properties.
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Restriction
In the discrete measure example, we can restrict to 2S
instead of B. It pretty much has the same property.
Remark: why we dont say restrict on S?Another restriction example: is concentrated on some
interval [a, b].
Construction of B[a, b]. Then we can restrict on this-field without loosing its mathematical properties.
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Restriction
In the discrete measure example, we can restrict to 2S
instead of B. It pretty much has the same property.
Remark: why we dont say restrict on S?Another restriction example: is concentrated on some
interval [a, b].
Construction of B[a, b]. Then we can restrict on this-field without loosing its mathematical properties.
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L-S measure on Rn
Sketch of construction
The analogy of intervals: rectangles.
Open rectangles, closed rectangles, semi-closed
rectangles: just need to know their two vertices. Samenotation: (a, b]
The smallest -field containing all rectangles: B(Rn).
A L-S measure on Rn is a measure : B(Rn) R such
that (I) < for each bounded rectangle I.
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L-S measure on Rn
Sketch of construction
The analogy of intervals: rectangles.
Open rectangles, closed rectangles, semi-closed
rectangles: just need to know their two vertices. Samenotation: (a, b]
The smallest -field containing all rectangles: B(Rn).
A L-S measure on Rn is a measure : B(Rn) R such
that (I) < for each bounded rectangle I.
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L-S measure on Rn
Sketch of construction
The analogy of intervals: rectangles.
Open rectangles, closed rectangles, semi-closed
rectangles: just need to know their two vertices. Samenotation: (a, b]
The smallest -field containing all rectangles: B(Rn).
A L-S measure on Rn is a measure : B(Rn) R such
that (I) < for each bounded rectangle I.
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L-S measure on Rn
Sketch of construction
The analogy of intervals: rectangles.
Open rectangles, closed rectangles, semi-closed
rectangles: just need to know their two vertices. Samenotation: (a, b]
The smallest -field containing all rectangles: B(Rn).
A L-S measure on Rn is a measure : B(Rn) R such
that (I) < for each bounded rectangle I.
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L-S Measure on Rn (II)
Its distribution function is defined to be F = ((, x]).
These distribution functions are also
non-decreasing. For a b, that is, a1 b1, a2 b2, . . ., wehave: F(a) F(b).Right-continuous. F is right continuous in all variables.
On the other hand, just as in the 1-dim case, for any
non-decreasing, right continuous function, there exist a
unique measure onB
(R
n
) corresponding with it.
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L-S Measure on Rn (II)
Its distribution function is defined to be F = ((, x]).
These distribution functions are also
non-decreasing. For a b, that is, a1 b1, a2 b2, . . ., wehave: F(a) F(b).Right-continuous. F is right continuous in all variables.
On the other hand, just as in the 1-dim case, for any
non-decreasing, right continuous function, there exist a
unique measure onB
(R
n
) corresponding with it.
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L-S Measure on Rn (II)
Its distribution function is defined to be F = ((, x]).
These distribution functions are also
non-decreasing. For a b, that is, a1 b1, a2 b2, . . ., wehave: F(a) F(b).Right-continuous. F is right continuous in all variables.
On the other hand, just as in the 1-dim case, for any
non-decreasing, right continuous function, there exist a
unique measure onB
(R
n
) corresponding with it.
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L-S Measure on Rn (II)
Its distribution function is defined to be F = ((, x]).
These distribution functions are also
non-decreasing. For a b, that is, a1 b1, a2 b2, . . ., wehave: F(a) F(b).Right-continuous. F is right continuous in all variables.
On the other hand, just as in the 1-dim case, for any
non-decreasing, right continuous function, there exist a
unique measure onB
(R
n
) corresponding with it.
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L-S Measure on Rn (II)
Its distribution function is defined to be F = ((, x]).
These distribution functions are also
non-decreasing. For a b, that is, a1 b1, a2 b2, . . ., wehave: F(a) F(b).Right-continuous. F is right continuous in all variables.
On the other hand, just as in the 1-dim case, for any
non-decreasing, right continuous function, there exist a
unique measure onB
(R
n
) corresponding with it.
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Measure of a finite rectangle
This is a main difference.
((a, b]) = F(b) F(a).
Draw a two dimensional example to illustrate this point.
Page 442-443 describes an elaborated way of measuring arectangle by means of the distribution function, but that
formula is not often used. The reason is that later we will
see that we can define density function for most common,
usefuldistributions, and there is a easy way to calculate
measure of a rectangle (or an arbitrary region for thatmatter) using integral.
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Measure of a finite rectangle
This is a main difference.
((a, b]) = F(b) F(a).
Draw a two dimensional example to illustrate this point.
Page 442-443 describes an elaborated way of measuring arectangle by means of the distribution function, but that
formula is not often used. The reason is that later we will
see that we can define density function for most common,
usefuldistributions, and there is a easy way to calculate
measure of a rectangle (or an arbitrary region for thatmatter) using integral.
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Measure of a finite rectangle
This is a main difference.
((a, b]) = F(b) F(a).
Draw a two dimensional example to illustrate this point.
Page 442-443 describes an elaborated way of measuring arectangle by means of the distribution function, but that
formula is not often used. The reason is that later we will
see that we can define density function for most common,
usefuldistributions, and there is a easy way to calculate
measure of a rectangle (or an arbitrary region for thatmatter) using integral.
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An example of measurable function
A probability example to show the motivation. Let
(,F, P) be a probability space. To be more specific, letssay = {HEAD, TAIL-a, TAIL-b},F = {, , {HEAD} , {TAIL-a, TAIL-b}},
({HEAD}) = 12 , ({TAIL-a, TAIL-b}) = 12 .
is a probability measure. Interpretation: probability of
seeing HEAD or TAIL (including a,b, types) is both 1/2.
Now define a function hon in this way: h(HEAD) = 1,
h(TAIL-a) = h(TAIL-b) = 1. (Such a function is sometimescalled a coding function).
Qiu, Lee BST 401
A l f bl f i
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An example of measurable function
A probability example to show the motivation. Let
(,F, P) be a probability space. To be more specific, letssay = {HEAD, TAIL-a, TAIL-b},F = {, , {HEAD} , {TAIL-a, TAIL-b}},
({HEAD}) = 12 , ({TAIL-a, TAIL-b}) = 12 .
is a probability measure. Interpretation: probability of
seeing HEAD or TAIL (including a,b, types) is both 1/2.
Now define a function hon in this way: h(HEAD) = 1,
h(TAIL-a) = h(TAIL-b) = 1. (Such a function is sometimescalled a coding function).
Qiu, Lee BST 401
A l f bl f ti
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An example of measurable function
A probability example to show the motivation. Let
(,F, P) be a probability space. To be more specific, letssay = {HEAD, TAIL-a, TAIL-b},F = {, , {HEAD} , {TAIL-a, TAIL-b}},
({HEAD}) = 12 , ({TAIL-a, TAIL-b}) = 12 .
is a probability measure. Interpretation: probability of
seeing HEAD or TAIL (including a,b, types) is both 1/2.
Now define a function hon in this way: h(HEAD) = 1,
h(TAIL-a) = h(TAIL-b) = 1. (Such a function is sometimescalled a coding function).
Qiu, Lee BST 401
A l f bl f ti (II)
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An example of measurable function (II)
This function codes HEAD, TAIL-a, TAIL-b into numbers,
which are much easier to process than plain English
descriptions!
One nice thing is, we can talk about P(h = 1) andP(h= 1) instead of (HEAD), ({TAIL-a, TAIL-b}).
hnot only maps arbitrary events into tangible numbers, but
also maps a measure defined on an arbitrary space to a
measure defined for numbers.
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A l f bl f ti (II)
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An example of measurable function (II)
This function codes HEAD, TAIL-a, TAIL-b into numbers,
which are much easier to process than plain English
descriptions!
One nice thing is, we can talk about P(h = 1) andP(h= 1) instead of (HEAD), ({TAIL-a, TAIL-b}).
hnot only maps arbitrary events into tangible numbers, but
also maps a measure defined on an arbitrary space to a
measure defined for numbers.
Qiu, Lee BST 401
An example of measurable function (II)
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An example of measurable function (II)
This function codes HEAD, TAIL-a, TAIL-b into numbers,
which are much easier to process than plain English
descriptions!
One nice thing is, we can talk about P(h = 1) andP(h= 1) instead of (HEAD), ({TAIL-a, TAIL-b}).
hnot only maps arbitrary events into tangible numbers, but
also maps a measure defined on an arbitrary space to a
measure defined for numbers.
Qiu, Lee BST 401
Another example (II)
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Another example (II)
More complex examples: measuring height of trees. :trees (descriptive). P: probability of the height of trees.
h : R maps every tree to a real number (ofcentimeters, or inches, etc).
In this example, we may want to estimate probabilities inthis form: P(a< h() b), i.e., , the probability of acertain range of height.
A notation: for any function h : R and for a set A R,denote h1(A) to be { |h() A}. Draw a diagram toshow this set.
By this notation, P(a< h() b) = P(h1((a, b])).
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Another example (II)
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Another example (II)
More complex examples: measuring height of trees. :trees (descriptive). P: probability of the height of trees.
h : R maps every tree to a real number (ofcentimeters, or inches, etc).
In this example, we may want to estimate probabilities inthis form: P(a< h() b), i.e., , the probability of acertain range of height.
A notation: for any function h : R and for a set A R,denote h1(A) to be { |h() A}. Draw a diagram toshow this set.
By this notation, P(a< h() b) = P(h1((a, b])).
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Another example (II)
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Another example (II)
More complex examples: measuring height of trees. :trees (descriptive). P: probability of the height of trees.
h : R maps every tree to a real number (ofcentimeters, or inches, etc).
In this example, we may want to estimate probabilities inthis form: P(a< h() b), i.e., , the probability of acertain range of height.
A notation: for any function h : R and for a set A R,denote h1(A) to be { |h() A}. Draw a diagram toshow this set.
By this notation, P(a< h() b) = P(h1((a, b])).
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Another example (II)
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Another example (II)
More complex examples: measuring height of trees. :trees (descriptive). P: probability of the height of trees.
h : R maps every tree to a real number (ofcentimeters, or inches, etc).
In this example, we may want to estimate probabilities inthis form: P(a< h() b), i.e., , the probability of acertain range of height.
A notation: for any function h : R and for a set A R,denote h1(A) to be { |h() A}. Draw a diagram toshow this set.
By this notation, P(a< h() b) = P(h1((a, b])).
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Non-measurable function
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Non-measurable function
Now let us define another function g on . g(HEAD) = 1,g(TAIL-a) = 0, g(TAIL-b) = 1.
This function is bad because P(g = 1) and P(g = 0) are
not well defined!Gambling interpretation: TAIL-a has certain probability that
is unmeasurable for us players. All we can observe are the
HEAD: lose one dollar; TAIL: most of time (TAIL-b) we win
one dollar, but sometimesthe result is canceled by the
casino (TAIL-a).
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Non-measurable function
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Non measurable function
Now let us define another function g on . g(HEAD) = 1,g(TAIL-a) = 0, g(TAIL-b) = 1.
This function is bad because P(g = 1) and P(g = 0) are
not well defined!Gambling interpretation: TAIL-a has certain probability that
is unmeasurable for us players. All we can observe are the
HEAD: lose one dollar; TAIL: most of time (TAIL-b) we win
one dollar, but sometimesthe result is canceled by the
casino (TAIL-a).
Qiu, Lee BST 401
Non-measurable function
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Non measurable function
Now let us define another function g on . g(HEAD) = 1,g(TAIL-a) = 0, g(TAIL-b) = 1.
This function is bad because P(g = 1) and P(g = 0) are
not well defined!Gambling interpretation: TAIL-a has certain probability that
is unmeasurable for us players. All we can observe are the
HEAD: lose one dollar; TAIL: most of time (TAIL-b) we win
one dollar, but sometimesthe result is canceled by the
casino (TAIL-a).
Qiu, Lee BST 401
Random variables and Borel-measurable functions
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Random variables and Borel measurable functions
Start with an arbitrary probability space: (,F, P).
A function h : R is called a random variable ifh1((a, b]) is always measurable, i.e., , P(a< h(x) b) isalways well defined.
From the Carathodory extension theorem, we know we
can extend intervals to Borel sets. In other words, if h is arandom variable, then for any Borel set B, h1(B) ismeasurable.
R can be extended to Rn. I.e., functions taking vector
values. These functions are called n-dimensional randomvectors.
Furthermore, if P is replaced by an arbitrary measure, h is
called an n-dimensional Borel-measurable function.
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Random variables and Borel-measurable functions
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Random variables and Borel measurable functions
Start with an arbitrary probability space: (,F, P).
A function h : R is called a random variable ifh1((a, b]) is always measurable, i.e., , P(a< h(x) b) isalways well defined.
From the Carathodory extension theorem, we know we
can extend intervals to Borel sets. In other words, if h is arandom variable, then for any Borel set B, h1(B) ismeasurable.
R can be extended to Rn. I.e., functions taking vector
values. These functions are called n-dimensional randomvectors.
Furthermore, if P is replaced by an arbitrary measure, h is
called an n-dimensional Borel-measurable function.
Qiu, Lee BST 401
Random variables and Borel-measurable functions
http://find/http://goback/ -
8/8/2019 Probability Theory Presentation 06
50/52
Random variables and Borel measurable functions
Start with an arbitrary probability space: (,F, P).
A function h : R is called a random variable ifh1((a, b]) is always measurable, i.e., , P(a< h(x) b) isalways well defined.
From the Carathodory extension theorem, we know we
can extend intervals to Borel sets. In other words, if h is arandom variable, then for any Borel set B, h1(B) ismeasurable.
R can be extended to Rn. I.e., functions taking vector
values. These functions are called n-dimensional randomvectors.
Furthermore, if P is replaced by an arbitrary measure, h is
called an n-dimensional Borel-measurable function.
Qiu, Lee BST 401
Random variables and Borel-measurable functions
http://find/http://goback/ -
8/8/2019 Probability Theory Presentation 06
51/52
a do a ab es a d o e easu ab e u ct o s
Start with an arbitrary probability space: (,F, P).
A function h : R is called a random variable ifh1((a, b]) is always measurable, i.e., , P(a< h(x) b) isalways well defined.
From the Carathodory extension theorem, we know we
can extend intervals to Borel sets. In other words, if h is arandom variable, then for any Borel set B, h1(B) ismeasurable.
R can be extended to Rn. I.e., functions taking vector
values. These functions are called n-dimensional randomvectors.
Furthermore, if P is replaced by an arbitrary measure, h is
called an n-dimensional Borel-measurable function.
Qiu, Lee BST 401
Random variables and Borel-measurable functions
http://find/http://goback/ -
8/8/2019 Probability Theory Presentation 06
52/52
Start with an arbitrary probability space: (,F, P).
A function h : R is called a random variable ifh1((a, b]) is always measurable, i.e., , P(a< h(x) b) isalways well defined.
From the Carathodory extension theorem, we know we
can extend intervals to Borel sets. In other words, if h is arandom variable, then for any Borel set B, h1(B) ismeasurable.
R can be extended to Rn. I.e., functions taking vector
values. These functions are called n-dimensional randomvectors.
Furthermore, if P is replaced by an arbitrary measure, h is
called an n-dimensional Borel-measurable function.
Qiu, Lee BST 401
http://find/http://goback/