Matheatical Methods, Appunti - random variables

The Modern Probability Theory

Modern probability theory is axiomatic. That means that the mathematical discipline of probability theory is developed from axioms, alinas (from axiom definition) fundamental basic principles accepted as true (without demonstration) and used as basis for the theory development. More precisely, that means first of all to define the elements to be studied within their basic relations (typically called primitives of the theory), secondly to state the axioms by which these relations are going to be governed and lastly develop the whole theory only being based on these few but well-posed elements (cf. Kolmogorov)

• Primitives

Sample space Ω: given a certain experiment, the set of all its possible outcomes. It may be countable or not. A set is countable iff it exist a bijection between it and ℕx where ℕx⊆ℕ+ and where ℕ+=ℕ∪{0}. Iff Ω is countable, it’s element are typically indicated as ω₁,ω₂,… (if not countable, then it’s continuous)

Event E: a subset of Ω, alias E⊆Ω.

From the definition of event is immediatly obtained that events and sets are isomorphic, result that can be exploited to illustrate the so called "Algebra of Events". Notice that two structures are said to be isomorphic iff they can be "mapped biunivocally onto eachother in a way that for each part of one structure can be univocally find a part of the other one that play an analogous role, and viceversa.

• Union, sum or: A∪B, A+B, A∨B
• Intersection, product, and: A∩B, A·B, A∧B
• Complement, negation, not: Ā
• Difference: A-B = A∩B̅
• Idempotence: A∪A= A∩A=A
• Associative laws: (A∪B)∪C = A∪(B∪C) and (A∩B)∩C = A∩(B∩C)
• Commutative laws: A∪B = B∪A and A∩B = B∩A
• Distributive laws: (A∩B)∪C = (A∪C)∩(B∪C) and (A∪B)∩C = (A∩B)∪(B∩C)
• Null event: A=∅ (alias empty set)
• Identity laws: A∪Ω=A, A∩Ω=Ω, A∩∅=∅ and A∪∅=A
• Complement Laws: A∪Ā=Ω, A∩Ā=∅, Ā=Ω, Ω̅=∅ and ∅̅=Ω
• DeMorgan Laws: A̅∪B̅=(A∩B)̅ and A̅∩B̅=(A∪B)̅

Def: A,B mutually exclusive iff A∩B=∅

Def: A,B exhaustive iff A∪B=Ω

Def of Partition: set of all exclusive and exhaustive events

The Modern Probability Theory

Modern probability theory is axiomatic. That means that the mathematical discipline of probability theory is developed from axioms, alms (from axiom definition) fundamental basic principles accepted as true (without demonstration) and used as basis for the theory development. More precisely, that means first of all to define the elements to be studied within their basic relations (typically called primitives of the theory), secondly to state the axioms by which these relations are going to be governed and lastly develop the whole theory only being based on these few but well-posed elements (cf. Kolmogorov)

• Primitives

Sample space Ω: given a certain experiment, the set of all its possible outcomes. It may be countable or not. A set is countable iff it exist a bijection between it and IN⁺ where IN⁺⊆IN⁺ and where IN⁺=IN∪{0}. Iff Ω is countable, it’s element are typically indicated as u₁,u₂,... (if not countable, then it's continuous)

Event E: a subset of Ω, alias E⊆Ω.

From the definition of event is we immediatly obtained that events and sets are isomorphic, result that can be exploited to illustrate the so called “Algebra of Events”. Notice that two structures are said to be isomorphic iff they can be mapped biunivocally onto eachother in a way that for each part of one structure can be univocally find a part of the other one that play on analogous role, and viceversa.

• Union, sum, or: A∪B, A+B, A∨B
• Intersection, product, and: A∩B, A・B, A∧B
• Complement, negation, not: Ā
• Difference: A-B = A∩B̄
• Idempotence: A∪A=A∩A=A
• Associative laws: (A∪B)∪C = A∪(B∪C) and (A∩B)∩C = A∩(B∩C)
• Commutative laws: A∪B=B∪A and A∩B=B∩A
• Distributive laws: (A∪B)∩C = (A∩B)∪(B∩C) and (A∩B)∪C = (A∪B)∩(B∪C)
• Null event: A=∅ (alias empty set)
• Identity laws: A∪Ω=Ω, A∩Ω=Ω, A∪∅=A and A∩∅=A
• Complement laws: A∪Ā=Ω, A∩Ā=∅, Ā=A, ∅̄=Ω and Ω̄=∅
• DeMorgan Laws: A∪B̄=Ā∩B̄ and A∩B̄=Ā∪B̄

Def: A,B mutually exclusive iff A∩B=∅Def: A,B exhaustive iff A∪B=ΩDef: Partition: set of all exclusive and exhaustive events

Def. of Field: a set ℰ of events closed w.r.t. negation and union, also they both holds A∈ℰ ∀A∈ℰ and A∪B∈ℰ ∀A,B∈ℰ (under the hypothesis of unions only of FINITE number of events).

Corollary: if ℰ is a field THEN they (also) holds A∩B∈ℰ and A-B∈ℰ ∀A,B∈ℰ

Def. of Borel Field: a field ℰ closed also when considering unions of infinite number of events.

Corollary: if ℰ is a Borel field THEN they (also) holds A∩B∈ℰ and A-B∈ℰ ∀A,B∈ℰ where the intersection is also closed when considering an infinite number of events.

Def. of probability: a real number associated to an event.

Axioms

Given a sample space Ω and a field ℰ over Ω, it exist a function that to each event of ℰ associates a real number (called to be its probability) for which they hold the Kolomogorov's axioms, alias the three properties:

• (E)≥0 ∀E∈ℰ
• (Ω)=1
• (A∪B)=(A)+(B) ∀A,B∈ℰ: A∩B=∅

Notice that the third axiom can be extended to union of an infinite number of events iff ℰ is a Borel Field, otherwise it can be extended only for a finite number of them.

Given the axioms, {ℰi:i=1,...} partition of Ω is called to be an "equally likely" iff (ℰi)=const. ∀i.

Given the axioms, we define as Probability Space the tuple (Ω,ℰ,).

Immediate Corollaries of the Axioms

• (E)=1-(E̅) ∀E∈ℰ

Proof: 1=(Ω)=(E∪E̅)=(E)+(E̅) => (E)=1-(E̅)

• (∅)=0

Proof: (Ω)=(∅̅)=1-(∅)=1-(∅)=1=>0=1-1=0

• (E)∈[0,1] ∀E∈ℰ

Proof: (E)=1-(E̅)∧(E)≥0=>1-(E̅)≥0 => (E)≥0 => (E)≤1

• A⊆B => (A)≤(B)

Proof: A⊆B => B≡(A∪(B∩Ā)) => (B)=(A)+(B∩Ā) ≥ (A)

(A∪B)=(A)+(B)-(A∩B) ∀A,B∈ℰ

Proof: A∪B=A∪(A∩B)∪Ā/B=(A∩B)∪(Ā∩B)

=> (A∪B)=(A)+((A∩B)∪(Ā∩B))

=> (A∪B)=(A)+(B)-(A∩B)

Can be extended to multi-events case:

(∪_i=1ⁿA_i)=∑_i=1ⁿ(A_i)-∑_i