Download Probability Space and Random Variables

Probability Space and Random
Variables
Measurable Space – σ-Algebra
Definition. (σ -Algebra)
A collection F of subsets of a set Ω is said to be a
σ -algebra in Ω if F has the following properties
(i) Ω ∈ F
(ii) A ∈ F ⇒ Ac ≡ Ω − A ∈ F
∞
S
(iii) A1 , A2 , . . . , An , . . . ∈ F ⇒
An ∈ F
n=1
(i), (ii): ∅ = Ωc ∈ F
An ∈ Ω ⇒
∞
T
n=1
An =
(ii)
∞
S
Acn
n=1
∈F ⇒
Acn ∈ F
(iii)
∞
S
n=1
Acn ∈ F ⇒(ii)
Probability Space and Random Variables – p.1/1
Measurable Space – σ-Algebra
If F is σ -algebra in Ω, then (Ω, F ) (or just Ω) is called
a measurable space, and A, B ∈ F are called
measurable sets in Ω
Probability Space and Random Variables – p.2/1
Measurable Space – σ-Algebra
Examples for σ -algebras in Ω = {1, 2, 3, 4, 5, 6}
F1 = Ω, ∅}
F2 = Ω, ∅, {1, 2} , {3, 4, 5, 6}
F
3 =
Ω, ∅, {1} , {2} , {1, 2} , {3, 4, 5, 6} , {1, 3, 4, 5, 6} ,
{2, 3, 4, 5, 6}
Examples for σ -algebras in Ω = R ≡ (−∞, ∞)
F1 = Ω, ∅}
F2 = Ω, ∅, (−∞, 0), [0, ∞)
F2 = Ω, ∅, {0}, (−∞, 0), (0, ∞), (−∞, 0], [0, ∞)
Probability Space and Random Variables – p.3/1
Measurable Space – σ(A)
Theorem If A is any collection of subsets of Ω, there
exists a smallest σ -algebra F ∗ in Ω such that A ⊂ F ∗ .
Definition F ∗ is called σ -algebra generated by A
and is denoted by σ(A).
Examples:
Ω = {1, 2, 3, 4, 5, 6}, A = {1, 2}
σ(A) = Ω, ∅, {1, 2} , {3, 4, 5, 6}
Ω = {1, 2, 3, 4, 5, 6}, A = {1} , {1, 2}
σ(A)
=
Ω, ∅, {1} , {2} , {1, 2} , {3, 4, 5, 6} , {1, 3, 4, 5, 6} ,
{2, 3, 4, 5, 6}
Probability Space and Random Variables – p.4/1
Measurable Space – σ(A)
Examples cont.:
Ω = (−10, 10], A = {1} , (0, 10]
σ(A) = Ω, ∅, {1} , (0, 10], (0, 1) ∪ (1, 10],
(−10, 0], (−10,1) ∪ (1, 10], (−10, 0] ∪ {1} ,
(0, 1) ∪ (1, 10]
Probability Space and Random Variables – p.5/1
Measurable Space – σ(A)
Procedure. Construct the σ -algebra F ∗ = {Fj }
generated from A = {Ai }
1: F ∗ ← Ω, ∅, Ai
2: repeat
3: F ∗ ← Fjc ∈
/ F ∗ (insert the missing current sets’
4:
5:
6:
7:
complements)
if there are no new sets inserted into F ∗ then
return F ∗
F ∗ ← Fj ∪ Fk ∈
/ F ∗ , j 6= k (insert the missing unions
of current sets)
until no change in iteration
return F ∗
Probability Space and Random Variables – p.6/1
Measurable Space – Borel σ-algebra
Let S be a topological space, the Borel σ -algebra on S ,
denoted by B(S), is the σ -algebra generated by the family
of open subsets of S , i.e.,
B(S) ≡ σ(open sets)
The member of B are called Borel sets of S .
Probability Space and Random Variables – p.7/1
Measurable Space – Measure Function
Let (Ω, F ) be a measurable space.
Definition. (Positive measure)
A ( (positive) measure) is a function µ
µ : F → [0, ∞]
which is countably additive, i.e., if {Ai } is a disjoint
countable collection of members of F , then
µ
∞
[
i=1
Ai =
∞
X
µ(Ai )
i=1
A measure space is the triplet (Ω, F , µ)
Probability Space and Random Variables – p.8/1
Measurable Space – Measure Function
Lemma. Let (Ω, F , µ) be a measure space then
µ(A ∪ B) ≤ µ(A) + µ(B), A, B ∈ F
S
P
∞
∞
µ
Ai ≤
µ(Ai ), Ai ∈ F
i=1
i=1
If µ(Ω) < ∞, then µ(A ∪ B) = µ(A) + µ(B) − µ(A ∩ B)
Probability Space and Random Variables – p.9/1
Measurable Space – Lebesgue Measure on R
n
Definition. (Lebesgue Measure) Let
(Ω, F ) = (Rn , B(Rn )). Let
n
o
A = x = [x1 , . . . , xn ]T ∈ Rn : αi ≤ xi < βi , i = 1, . . . , n
then the Lebesgue measure of A denoted by m(A) is
defined as
∞
Y
m(A) = vol(A) ≡
(βi − αi )
i=1
if Ω = R what is the Lebesgue measure of A = [α, β)?
Answer: its length m(A) = β − α
Probability Space and Random Variables – p.10/1
Probability Space – Probability Function
Definition. (Probability Measure)
Let Ω denotes the sample space and F a collection
of events assumed to be a σ -algebra of events.
Then, a probability measure (function) is measure
such that P (Ω) = 1, i.e.,
P : F → [0, 1]
which leads to the probability axioms:
(i) P (A) ≥ 0 ∀A ∈ F
(ii) P (Ω) = 1
(iii) Ai ∈ F , Ai ∩ Aj = ∅ ∀i 6= j; i, j = 1, 2, . . .
∞
∞
P
S
P
Ai =
P (Ai )
i=1
i=1
Probability Space and Random Variables – p.11/1
Probability Space – Definition
Definition. (Probability Space)
A probability space is the triplet (Ω, F , P ) where Ω is the
sample space, F is a σ -algebra of events, and P (·) is a
probability measure (function) with domain Ω and range
[0, 1].
Probability Space and Random Variables – p.12/1
Random Variable – Definition
Definition. (Random variable)
Let (Ω, F , P ) be a probability space. A random variable ,
denoted by X or X(·) is a function
X :Ω→R
and must satisfies
Ax = {ω : X(w) ≤ x} ∈ F
Probability Space and Random Variables – p.13/1
Random Variable – Examples
Consider the experiment of tossing a dice:
Ω = {ω : ω = 1, 2, . . . , 6}
X1 (ω) = ω
(
X2 (ω) =
0
1
ω ∈ {1, 3, 5}
ω ∈ {2, 4, 6}
Consider the experiment of tossing a dice twice:
Ω = {ω = (i, j) : i, j = 1, 2, . . . , 6}
X1 (ω) = i + j
X2 (ω) = |i − j|
Probability Space and Random Variables – p.14/1
Random Variable – Definitions
Definition. (Cumulative distribution function)
The (cumulative) distribution function (cdf) of a random
variable X , denoted by FX (· · · ) , is defined to be
FX : R → [0, 1]
which satisfies
FX (x) = P (X ≤ x) = P {ω : X(w) ≤ x}
Probability Space and Random Variables – p.15/1
Random Variable – cdf
Examples. (Cumulative distribution function)
Consider the experiment of tossing a dice once


x<1
0
FX (x) = ⌊x⌋
1≤x≤6
6


1
x>6
which satisfies
FX (x) = P (X ≤ x) = P {ω : X(w) ≤ x}
Probability Space and Random Variables – p.16/1
Random Variable – cdf
Examples. (Cumulative distribution function)
Probability Space and Random Variables – p.17/1