probability theory presentation 06

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BST 401 Probability Theory

Xing Qiu Ha Youn Lee

Department of Biostatistics and Computational BiologyUniversity of Rochester

September 21, 2009

Qiu, Lee BST 401
http://find/http://goback/


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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 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










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Generalizations










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Generalizations










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Generalizations










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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.





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Discrete measure

Page 26.





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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




interval [a, b].


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Restriction




interval [a, b].


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Restriction




interval [a, b].


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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








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L-S measure on Rn








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L-S measure on Rn








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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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unique measure onB

(R

n


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unique measure onB

(R

n


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unique measure onB

(R

n


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unique measure onB

(R

n


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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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((a, b]) = F(b) F(a).







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((a, b]) = F(b) F(a).







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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).

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A l f bl f i


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({HEAD}) = 12 , ({TAIL-a, TAIL-b}) = 12 .





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A l f bl f ti


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({HEAD}) = 12 , ({TAIL-a, TAIL-b}) = 12 .





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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.

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A l f bl f ti (II)


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descriptions!





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descriptions!





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Another example (II)


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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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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







casino (TAIL-a).

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Non measurable function







casino (TAIL-a).

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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.

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a do a ab es a d o e easu ab e u ct o s









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probability theory presentation 06

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