probability theory presentation 04

8/8/2019 Probability Theory Presentation 04

1/34

BST 401 Probability Theory

Xing Qiu Ha Youn Lee

Department of Biostatistics and Computational BiologyUniversity of Rochester

September 14, 2009

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


2/34

Outline

1 Measures

Qiu, Lee BST 401


3/34

Countable additivity of a set function

Let () be an extended real-valued set function defined on a

-field F, i.e.,

It takes sets A in F as input variables.

It maps these sets into extended real numbers, e.g., real

numbers plus .

It is said to befinitely additive if: For finite collection of disjoint sets

A1, . . . ,AN in F, N

n An

=

n(An).

Countably additive (-additive) if: For countably infinite

collection of disjoint sets A1, . . . ,An, . . . in F,

n

An

=

n

(An). (1)

Qiu, Lee BST 401


4/34



-field F, i.e.,



numbers plus .


A1, . . . ,AN in F, N

n An

=

n(An).



n

An

=

n

(An). (1)

Qiu, Lee BST 401


5/34



-field F, i.e.,



numbers plus .


A1, . . . ,AN in F, N

n An

=

n(An).



n

An

=

n

(An). (1)

Qiu, Lee BST 401


6/34



-field F, i.e.,



numbers plus .


A1, . . . ,AN in F, N

n An

=

n(An).



n

An

=

n

(An). (1)

Qiu, Lee BST 401


7/34

Basic definition

A measure on a -field F is a function (A) such that

It is non-negative: (A) 0.

Measure of an empty set is always zero: () = 0.(Remark: The reverse is not true.)

It is countably additive.

Exercise: finite/countable additivity are about commuting union

and addition of disjointsets. What happens if A1,A2, . . . are not

disjoint?

Qiu, Lee BST 401


8/34

Basic definition







disjoint?

Qiu, Lee BST 401


9/34

Basic definition







disjoint?

Qiu, Lee BST 401


10/34

Finite/Probability Measure

If () is finite, is called a finite measure.

Let 2(A) = C1(A). 1, 2 share a lot of mathematicalproperties.

As a special case, for every finite measure , we can

definite (A) = 1()(A) so that

() = 1.

If () = 1, is called a probability measure.

In other words, essentially every finite measure is

equivalent to a probability measure.

By convention, we define a measure/probability space to be a

triple (,F, ). Where is the whole space, F is the -fieldwith as its whole space, and is a measure defined on F.

Qiu, Lee BST 401


11/34






() = 1.






Qiu, Lee BST 401


12/34






() = 1.






Qiu, Lee BST 401


13/34






() = 1.






Qiu, Lee BST 401


14/34






() = 1.






Qiu, Lee BST 401


15/34

About

The term of a probability space is a set of all outcomes,and are not necessarily numbers.

For example, we sometimes denote the set of two possible

outcomes of a Bernoulli distribution as = {H,T},represents Head and Tail.

In fact you can have defined as the set of humans suchasall students who are taking BST401, a -algebra

containing all subsets of you, and define a probability to

quantify the probability that one of you guys in such asubset will one day win the Nobel prize.

Qiu, Lee BST 401


16/34

About







Qiu, Lee BST 401


17/34

About







Qiu, Lee BST 401


18/34

Examples

General definition of a counting measure: for any , iswell defined on the largest -field 2.

Counting measures on a finite set, Z, and Q.

Formally prove that every counting measure is a valid

measure.Discrete probabilities: K unique outcomes:

= {1, . . . , K}, p1, . . . , pK,K

k=1 pk = 1,(A) =

kA

pk Here K can be countably infinite.

Bernoulli distribution. (fair coin/non-fair coin).

Binomial distribution. = {0,1, . . . ,N}, F = 2pk =

Nk

pk(1 p)Nk.

Poisson distribution. = Z+ = {0,1,2, . . .}, F = 2Z+

,

pk = e

k

k! .

Qiu, Lee BST 401


19/34

Examples





= {1, . . . , K}, p1, . . . , pK,K

k=1 pk = 1,(A) =

kA




Nk

pk(1 p)Nk.


,

pk = e

k

k! .

Qiu, Lee BST 401


20/34

Examples





= {1, . . . , K}, p1, . . . , pK,K

k=1 pk = 1,(A) =

kA




Nk

pk(1 p)Nk.


,

pk = e

k

k! .

Qiu, Lee BST 401


21/34

Examples





= {1, . . . , K}, p1, . . . , pK,K

k=1 pk = 1,(A) =

kA




Nk

pk(1 p)Nk.


,

pk = e

k

k! .

Qiu, Lee BST 401


22/34

Examples





= {1, . . . , K}, p1, . . . , pK,K

k=1 pk = 1,(A) =

kA




Nk

pk(1 p)Nk.


,

pk = e

k

k! .

Qiu, Lee BST 401


23/34

Examples





= {1, . . . , K}, p1, . . . , pK,K

k=1 pk = 1,(A) =

kA




Nk

pk(1 p)Nk.


,

pk = e

k

k! .

Qiu, Lee BST 401


24/34

Examples





= {1, . . . , K}, p1, . . . , pK,K

k=1 pk = 1,(A) =

kA




Nk

pk(1 p)Nk.


,

pk = e

k

k! .

Qiu, Lee BST 401


25/34

Basic Properties of a Measure

Monotonicity. If A B, then (A) (B).

Subadditivity. If A iA

i, then (A)

i(A

i). As a

special case, (iAi)

i(Ai).

Continuity from below. Ai A = (Ai) (A).

Continuity from above. Ai A = (Ai) (A).

Qiu, Lee BST 401


26/34




i, then (A)

i(A

i). As a

special case, (iAi)

i(Ai).



Qiu, Lee BST 401


27/34




i, then (A)

i(A

i). As a

special case, (iAi)

i(Ai).



Qiu, Lee BST 401


28/34




i, then (A)

i(A

i). As a

special case, (iAi)

i(Ai).



Qiu, Lee BST 401


29/34

The Borel -fields

Borel measure on R: = R, B, (A) = the length of A.Where the Borel -field is defined to be the smallest -field

containing all open intervals (a,b). (or all closed intervals[a,b], or all intervals of form (a, b].)

First step: open sets. More details will be discussed later.Terminology: the smallest -field containing a collection of

sets S is called a) the minimal-field over S; b) the

-field generated by S.

B is a -field generated by the collection of open/closed

intervals/sets.

Not every subset of R is a Borel set! Checkout the Vitali set

from Wikipedia.

Qiu, Lee BST 401


30/34

The Borel -fields







intervals/sets.


from Wikipedia.

Qiu, Lee BST 401


31/34

The Borel -fields







intervals/sets.


from Wikipedia.

Qiu, Lee BST 401


32/34

The Borel -fields







intervals/sets.


from Wikipedia.

Qiu, Lee BST 401


33/34

The Borel -fields







intervals/sets.


from Wikipedia.

Qiu, Lee BST 401


34/34

The Borel -fields







intervals/sets.


from Wikipedia.

Qiu, Lee BST 401

probability theory presentation 04

Documents