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Probability Spaces Will Perkins January 3, 2013 Sigma Fields - - PowerPoint PPT Presentation
Probability Spaces Will Perkins January 3, 2013 Sigma Fields - - PowerPoint PPT Presentation
Probability Spaces Will Perkins January 3, 2013 Sigma Fields Definition A sigma-field ( -field) F is a collection (family) of subsets of a space satisfying: 1 F 2 If A F , then A c F 3 If A 1 , A 2 , F , then
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Sigma Fields
Definition A sigma-field (σ-field) F is a collection (family) of subsets of a space Ω satisfying:
1 Ω ∈ F 2 If A ∈ F, then Ac ∈ F 3 If A1, A2, · · · ∈ F, then ∞ i=1 Ai ∈ F
Examples:
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Sigma Fields
Definition A sigma-field (σ-field) F is a collection (family) of subsets of a space Ω satisfying:
1 Ω ∈ F 2 If A ∈ F, then Ac ∈ F 3 If A1, A2, · · · ∈ F, then ∞ i=1 Ai ∈ F
Examples: The set of all subsets of Ω is a σ-field
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Sigma Fields
Definition A sigma-field (σ-field) F is a collection (family) of subsets of a space Ω satisfying:
1 Ω ∈ F 2 If A ∈ F, then Ac ∈ F 3 If A1, A2, · · · ∈ F, then ∞ i=1 Ai ∈ F
Examples: The set of all subsets of Ω is a σ-field F = {Ω, ∅}
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Sigma Fields
Definition A sigma-field (σ-field) F is a collection (family) of subsets of a space Ω satisfying:
1 Ω ∈ F 2 If A ∈ F, then Ac ∈ F 3 If A1, A2, · · · ∈ F, then ∞ i=1 Ai ∈ F
Examples: The set of all subsets of Ω is a σ-field F = {Ω, ∅} F = {Ω, ∅, A, Ac}
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The Borel Sigma Field
Proposition Let F be any collection of subsets of Ω. Then there is a smallest sigma-field, σ(F) that contains F.
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The Borel Sigma Field
Proposition Let F be any collection of subsets of Ω. Then there is a smallest sigma-field, σ(F) that contains F. Proof: ?
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The Borel Sigma Field
Proposition Let F be any collection of subsets of Ω. Then there is a smallest sigma-field, σ(F) that contains F. Proof: ? Definition Let Ω be a metric space and F the collection of all open subsets of Ω. Then the Borel σ-field is σ(F)
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Probability
Definition Let F be a σ-field on Ω. Then P : F → [0, 1] is a probability measure if:
1 P(Ω) = 1 2 For any A1, A2, · · · ∈ F such that Ai ∩ Aj = ∅ for all i = j,
P ∞
- i=1
Ai
- =
∞
- i=1
P(Ai)
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Probability Space
Definition A probability space is a triple (Ω, F, P) of a space, a σ-field, and a probability function.
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Examples
1 Coin Flip: Ω = {H, T}, F = {Ω, ∅, {H}, {T}},
P(H) = 1/2, P(T) = 1/2.
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Examples
1 Coin Flip: Ω = {H, T}, F = {Ω, ∅, {H}, {T}},
P(H) = 1/2, P(T) = 1/2.
2 Two coin flips: Ω = {HH, HT, TH, TT}, F = { all subsets },
P(HH) = 1/4, . . .
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Examples
1 Coin Flip: Ω = {H, T}, F = {Ω, ∅, {H}, {T}},
P(H) = 1/2, P(T) = 1/2.
2 Two coin flips: Ω = {HH, HT, TH, TT}, F = { all subsets },
P(HH) = 1/4, . . .
3 Pick a uniform random number from [0, 1]: Ω = [0, 1]. F is
the Lebesgue or Borel σ-field. P is Lebesgue measure (i.e., P([a, b]) = b − a, the length of the interval).
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Language: Probability vs. Set Theory
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Language: Probability vs. Set Theory
1 Ω is the sample space. Any ω ∈ Ω is an outcome. The set of
- utcomes must describe the experiment exhaustively and
exclusively.
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Language: Probability vs. Set Theory
1 Ω is the sample space. Any ω ∈ Ω is an outcome. The set of
- utcomes must describe the experiment exhaustively and
exclusively.
2 An event is a set of outcomes that is in the σ-field, E ∈ F.
Anything you want to ask the probability of must be an event in the σ-field.
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Language: Probability vs. Set Theory
1 Ω is the sample space. Any ω ∈ Ω is an outcome. The set of
- utcomes must describe the experiment exhaustively and
exclusively.
2 An event is a set of outcomes that is in the σ-field, E ∈ F.
Anything you want to ask the probability of must be an event in the σ-field.
3 Intersection: A ∩ B is the event that A and B happen.
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Language: Probability vs. Set Theory
1 Ω is the sample space. Any ω ∈ Ω is an outcome. The set of
- utcomes must describe the experiment exhaustively and
exclusively.
2 An event is a set of outcomes that is in the σ-field, E ∈ F.
Anything you want to ask the probability of must be an event in the σ-field.
3 Intersection: A ∩ B is the event that A and B happen. 4 Union: A ∪ B is the event that A or B happens.
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Language: Probability vs. Set Theory
1 Ω is the sample space. Any ω ∈ Ω is an outcome. The set of
- utcomes must describe the experiment exhaustively and
exclusively.
2 An event is a set of outcomes that is in the σ-field, E ∈ F.
Anything you want to ask the probability of must be an event in the σ-field.
3 Intersection: A ∩ B is the event that A and B happen. 4 Union: A ∪ B is the event that A or B happens. 5 Complement: Ac is the event that A does not happen.
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Properties of a probability function
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Properties of a probability function
1 P(Ω) = 1, P(∅) = 0.
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Properties of a probability function
1 P(Ω) = 1, P(∅) = 0. 2 Monotonicity: If A ⊆ B, then P(A) ≤ P(B)
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Properties of a probability function
1 P(Ω) = 1, P(∅) = 0. 2 Monotonicity: If A ⊆ B, then P(A) ≤ P(B) 3 P(A ∪ B) = P(A) + P(B) − P(A ∩ B) (inclusion / exclusion)
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Properties of a probability function
1 P(Ω) = 1, P(∅) = 0. 2 Monotonicity: If A ⊆ B, then P(A) ≤ P(B) 3 P(A ∪ B) = P(A) + P(B) − P(A ∩ B) (inclusion / exclusion) 4 Let A1 ⊆ A2 ⊆ · · · , and let A = ∞ i=1 Ai. Then
P(A) = lim
n→∞ P(An)
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Properties of a probability function
1 P(Ω) = 1, P(∅) = 0. 2 Monotonicity: If A ⊆ B, then P(A) ≤ P(B) 3 P(A ∪ B) = P(A) + P(B) − P(A ∩ B) (inclusion / exclusion) 4 Let A1 ⊆ A2 ⊆ · · · , and let A = ∞ i=1 Ai. Then
P(A) = lim
n→∞ P(An) 5 Let A1 ⊇ A2 ⊇ · · · , and let A = ∞ i=1 Ai. Then
P(A) = lim
n→∞ P(An)
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