Lecture 1: Probability Review
Esther Frostig
The University of Haifa
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Random experiment: An experiment with with uncertain
outcomes.
1. throwing a dice
2. toss a coin twice.
3. toss a coin infintly many times.
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Sample space: Ω-The set of all possible outcomes of a
random experiment.
1. throwing a dice: Ω = {1, 2, 3, 4, 5, 6}.
2. toss a coin twice. Ω = {HH, HT , TH, TT }
3. toss a coin infintly many times.
Ω = {all infinite sequences of H and T}
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Evnt-A subset of Ω
1. In 1 Let A-the result is even. A = {2, 4, 6}
2. In 2. Let B-H in the first toss, B = {HH, HT }.
3. In 3 let C the first toss is H.
C = all sequences begin with H.
1. w ∈ A if the point (or result w) is in A. In 1. Let
A = {2, 4, 6}, then 2 ∈ A. If the die is tossed and "2"
appears we say that A occurred.
2. Let A and B be events (in Ω). A ⊂ B (B includes A) if w ∈ A
implied that w ∈ B. (if A occurred so is B).
3. A ⊂ B and B ⊂ A then A = B.
4. Ac or Ā : Ac = {w : w ∈
/ A}.
5. Φ = Ωc empty set.
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Let A and B be subsets of ω the union of A and B denoted
by A ∪ B is the set of elements that belong to A or B (or
both)
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Let A1 , A2 , · · · , An are sets in Ω then ∪ni=1 Ai is the sets of
outcome that are in at least one of the sets
A1 , A2 , · · · , An .similarily define ∪∞
i=1 Ai .
Ex.toss a coin twice. Ω = {HH, HT , TH, TT }. Let Ai be the
event H in the i toss. Then
A1 ∪ A2 = {HH, HT , TH},A1 ∩ A2 = {HH}.
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Let A and B be subsets of ω the intersection of A and B
denoted by A ∩ B is the set of elements that belong to both
A and B
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Let A1 , A2 , · · · , An ba sets in Ω then ∪ni=1 Ai is the set of
outcome that are at leasr in one of the sets A1 , A2 , · · · , An .
∩∞
i=1 Ai is the set of outcomes that are in all the sets
Ai , i = 1, · · · , n.
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A and B are disjoint if A ∩ B = Φ
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Let Ai ,B in Ω
B ∩ (∪i Ai ) = ∪i (B ∩ Ai )
B ∪ (∩i Ai ) = ∩i (B ∪ Ai )
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De Morgan Rules
(∪i Ai )c = ∩i Aci
(∩i Ai )c = ∪i Aci
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σ-field- A collection of subset of a non-empty set Ω is
called a σ-field, denoted by F if it satisfies the following
coniditions:
1. Φ ∈ F.
2. if A ∈ F so is Ac .
3. Whenever a sequence of sets A1 , A2 , · · · ∈ F then
∪∞
i=1 Ai ∈ F
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1. {Φ, Ω}-called trivial σ field.
2. For A ⊂ Ω, F = {Φ, Ω, A, Ac }-σ field generated by A.
3. When
Ω = {HH, HT , TH, TT },
(1)
F = {Φ, Ω, {HH}, {HT }, {TH}, {TT }, {TT , TH},
{HH, HT }, {HH, TH}, {HH, TT }, {TH, HT }, {TT , HT },
{HH, HT , TH}, {HH, HT , TT }, {HH, TH, TT }, {TH, HT , TT }}
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Borel σ-field Let Ω = [0, 1]. Consider the σ algebra
containing all closed interval (i.e. {[c, d] : 0 ≤ c < d ≤ 1}).
Then add what is necessary to form a σ-field. Clearly, the
σ field contains also all the open intervals since :
∞
[
n=1
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[a +
1
1
, b − ] = (a, b)
n
n
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Let F1 , and F2 two σ fiels defined on the same sample
space. Assume that F1 ⊂ F2 meaning that any event in F1
is also in F2 , then we say that F1 is smaller than F2 .
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Let A be a colection of subsets of Ω. The σ -field generated
by A is the smallest σ-field containing all the sets in A.
Example
In (1) Let A = {{HH, HT , TH}} Then the smallst σ-field
containing it is {Φ, Ω, {HH, HT , TH}, {TT }}
Definition
Let Ω be a nonempty set, and let F be a σ-field of subsets of Ω.
A probability measure P on (Ω, F) is a function P : F → [0, 1]
satisfying:
1. P(Ω) = 1
2. whenever A1 , A2 , · · · is a sequence of disjount sets in F,
(i.e. Ai ∩ Aj = Φ, i 6= j) then
!
∞
∞
[
X
P
Ai =
P(Ai )
i=1
i=1
1. property 2 holds for a finite union of disjoint sets: Let
A1 · · · , An disjoint sets. Define Aj = Φ for j = n + 1, cdots.
Then
!
!
∞
∞
n
n
[
X
X
[
Ai = P
Ai =
P(Ai ) =
P(Ai )
P
i=1
i=1
i=1
i=1
2. P(Ac ) = 1 − P(A).
3. Let B/A = B ∩ Ac . If A ⊂ B then
P(B) = P(A) + P(B/A) ≥ P(A)
4. P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
5. Generalization of 3: Let A1 , · · · , An be events then
P (∪ni=1 Ai )
=
n
X
i=1
+
X
i1 <i2 <i3
P(Ai ) −
X
P(Ai1 ∩ Ai2 )
i1 <i2
P(Ai1 ∩ Ai2 ∩ Ai3 ) + · · · + (−1)n+1 P(A1 ∩ · · · ∩ An )
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When Ω is finite e.g. tossig a coin twice,
Ω = {HH, HT , TH, TT }. then it is natural to take the σ field
as the colection of all subsets of Ω as in (2). In this case it
is enough to assign probability to each result in Ω.
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Increasing and decreasing sets.
Definition
A sequence of sets A1 , A2 , · · · is increasing if A1 ⊆ A2 ⊆ · · · .
For an increasig sequence define
A=
∞
[
i=1
Ai = lim Ai
i→∞
A sequence of sets B1 , B2 , · · · is decreasing if B1 ⊇ B2 ⊇ · · · .
For an decreasig sequence define
B=
∞
\
i=1
Bi = lim Bi
i→∞
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Increaing sequence: Ai = [1/i, 1 − 1/i], i = 1, 2, · · · .
A=
∞
[
i=1
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Ai = lim Ai = (0, 1)
i→∞
Decreaing sequence: Bi = [−1/i, 1 + 1/i], i = 1, 2, · · · .
B=
∞
\
i=1
Bi = lim Bi = [0, 1]
i→∞
Lemma
Let A1 ⊆ A2 ⊆ · · · be an increasing sequence. Then
P(
∞
[
Ai ) = P(limAi ) = lim P(Ai )
i=1
Proof.
Define D1 = A1 , D2 = A2 ∩ Ac1 , Dn+1 = An+1 ∩ Acn , etc.
P(Di ) = P(Ai ∩ Aci−1 ) = P(Ai ) − P(Ai ∩ Ai−1 ) = P(Ai ) − P(Ai−1 )
S
S
a. Dj are
b. An =P ni=1 Di . c. A = lim An = ∞
i=1 Di . d.
Pdisjoint.
n
P(A) = ∞
P(D
)
=
lim
P(D
)
=
lim
P(A
)
n
i
i
i=1
i=1
Lemma
Let B1 ⊆ B2 ⊆ · · · be a decreasing sequence. Then
P(∩∞
i=1 Bi ) = P(limBi ) = lim P(Bi )
Proof.
∞
c
Bic increasing. P(B) = P(∩∞
i=1 Bi ) = 1 − P((∩i=1 Bi ) ) =
c
c
c
1 − P(∪∞
i=1 Bi ) = 1 − limP(Bi ) = lim(1 − P(Bi )) = limP(Bi )
Theorem
Let Ω = [0, 1] and consider the σ-field generated by all the open
subset of [0, 1]. For 0 ≤ a < b ≤ 1 define
µ(a, b) = b − a
(2)
,
then there is a unique probability measure µ on the Borel set in
[0, 1] such that (2 ) holds for all 0 ≤ a < b ≤ 1.