Outline Outline Definitions Definitions Set Operations Set - - PowerPoint PPT Presentation

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• G. Ahmadi

ME 529 - Stochastics

• G. Ahmadi

ME 529 - Stochastics

Outline Outline

• Definitions

Definitions

• Set Operations

Set Operations

• Probability Space

Probability Space

• Borel

Borel Field Field

• Probability Experiment

Probability Experiment

• G. Ahmadi

ME 529 - Stochastics

Set is a collection of elements Set is a collection of elements Subset Subset b b of

• f a

a (all element of

(all element of b b are element of are element of a a) )

Space Space S S – – largest set largest set Null Set Null Set O O – – empty set empty set

S a b c S a b c

• G. Ahmadi

ME 529 - Stochastics

a b ⊂

If If

b c ⊂ a c ⊂

& &

b a = b a ⊂ a b ⊂

iff iff & &

Equality Equality

Subset: Subset: b

b is a subset of is a subset of a a if all element of if all element of b b are element of are element of a a) )

SLIDE 2

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• G. Ahmadi

ME 529 - Stochastics

a b b a ∪ = ∪

( ) ( )

c b a c b a c b a ∪ ∪ = ∪ ∪ = ∪ ∪

a a a = ∪

a a = ∪0

S S a = ∪

Elements of union of Elements of union of sets sets a a and and b b are are elements of elements of a a or

• r b

b or both

• r both
• G. Ahmadi

ME 529 - Stochastics

a b b a ∩ = ∩

a a a = ∩

0 = ∩ a

a S a = ∩

( ) ( )

c b a c b a c b a ∩ ∩ = ∩ ∩ = ∩ ∩

S S a = ∪

a b ⊂

If If

b a b = ∩

( ) ( ) ( )

c a b a c b a ∩ ∪ ∩ = ∪ ∩

Elements of intersection of Elements of intersection of sets sets a a and and b b are elements of both are elements of both a a and and b. b.

• G. Ahmadi

ME 529 - Stochastics

Mutually exclusive sets have no Mutually exclusive sets have no common element common element. .

= ∩b a

j i for a a

j i

≠ = ∩

Sets Sets a a1 ,

1 , a

a2

2 … are mutually exclusive if

… are mutually exclusive if

• G. Ahmadi

ME 529 - Stochastics

= S

If If

a b ⊂

S a a = ∪

= ∩ a a S = a b ⊃ Elements of complement of set Elements of complement of set a a are are elements of elements of S S which are not in which are not in a. a.

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• G. Ahmadi

ME 529 - Stochastics

b a b a ∩ = ∪

b a b a ∪ = ∩

• G. Ahmadi

ME 529 - Stochastics

a S a − = b a a b a b a ∩ − = ∩ = − = − ∪ a a a

( )

a a a a = ∪ −

( ) ( )

b a b a a ∩ ∪ − =

Elements of Elements of a a -

• b

b are elements of are elements of a a that that are not in are not in b. b.

• G. Ahmadi

ME 529 - Stochastics

Summary Summary

• G. Ahmadi

ME 529 - Stochastics

By an experiment By an experiment ℑ ℑ we mean a set (space) we mean a set (space) S S

• f outcomes
• f outcomes ξ

ξ. Elements of . Elements of S S are outcomes are outcomes

• r
• r elementary events

elementary events. . S S is a probability is a probability (sample) space. Subsets of (sample) space. Subsets of S S are called are called events

• events. Space

. Space S S is the is the sure (certain) sure (certain) event event. . Empty set Empty set O O is the is the impossible event impossible event. . Random Experiment Random Experiment ℑ ℑ Mutually Exclusive Events Mutually Exclusive Events

= ∩b a

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• G. Ahmadi

ME 529 - Stochastics

Axioms of Probability Axioms of Probability

If If

( ) ( ) ( )

b P a P b a P ) iii + = ∪

= ∩b a

( )

≥ a P ) i

( )

1 = S P ) ii

To each event To each event a, a, a measure a measure P(a P(a) ) is assigned is assigned subject to the following axioms subject to the following axioms

• G. Ahmadi

ME 529 - Stochastics

Corollaries Corollaries

( )

( ) 1

1 ≤ − = a P a P

( )

0 = P

If If

( ) ( )

( )

( )

b P b a P b P a P ≥ ∩ + =

a b ⊂

If If

( ) ( ) ( ) ( )

b a P b P a P b a P ∩ − + = ∪

≠ ∩b a

• G. Ahmadi

ME 529 - Stochastics

If If

F b∈

F a∈ If If

F b a ∈ ∪ F a∈ F a∈

& & If If

F b∈

F b a ∈ ∩ F a∈

& &

F b a ∈ − F S∈

F ∈

Also Also & &

Field F is a nonempty class of sets such that Field F is a nonempty class of sets such that

Corollaries Corollaries

• G. Ahmadi

ME 529 - Stochastics

If a field has the property that if the If a field has the property that if the sets sets a a1

1, a

, a2

2, …, a

, …, an

n,…

,… belong to it then belong to it then so does the set so does the set a a1

1 ∪

∪ a a2

2 ∪

∪ a a3

3 ∪

∪ … … ∪ ∪ a an

n ∪

∪ …, …, then the field is called a then the field is called a Borel Borel field. Note that the class of all

• field. Note that the class of all

subsets of subsets of S S is a is a Borel Borel field. field.

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• G. Ahmadi

ME 529 - Stochastics

Probability Experiment Probability Experiment ℑ ℑ: (S, F, P) : (S, F, P)

• 1. Set
• 1. Set S

S of outcomes

• f outcomes ξ

ξ; this set is called space ; this set is called space

• r sure (certain) event
• r sure (certain) event

2.

• 2. Borel

Borel field field F F consisting of certain subsets consisting of certain subsets

• f
• f S

S called events called events

• 3. Measure
• 3. Measure P(a

P(a) ) assigned to every event assigned to every event a a; ; this measure is called probability of this measure is called probability of event event a a, satisfies axioms 1 to 3 , satisfies axioms 1 to 3

• G. Ahmadi

ME 529 - Stochastics

• Example. Probability Experiment
• Example. Probability Experiment
• f Tossing a Coin,
• f Tossing a Coin, ℑ

ℑ: (S, F, P) : (S, F, P)

{ } {} { }

t h t h F , , , , :

{ }

t h S , =

( )

p h P =

{}

q t P = 1 = + q p

• G. Ahmadi

ME 529 - Stochastics

Concluding Remarks Concluding Remarks

• Definitions

Definitions

• Set Operations

Set Operations

• Probability Space

Probability Space

• Borel

Borel Field Field

• Probability Experiment

Probability Experiment

• G. Ahmadi

ME 529 - Stochastics