Basic Probability
Theory
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Continuous case. Kolmogorov’s approach
Definition1.
Let Ω be a set (Ω=S, sample space) and Σ be a family of subsets
of Ω.
Then Σ is the σ-field if:
Ω∈Σ
A∈ Σ ⇒ Ac ∈ Σ
A1, A 2 , A n ,... ∈ F ⇒
∞
U Ai ∈ Σ
i =1
Remarks. If Σ is a σ-field then
The pair
Ω, Σ
A1, A 2 , A n ,... ∈ Σ ⇒
∞
I Ai ∈ Σ
i =1
constitutes the measurable space.
Definition 2.
Let Ω, Σ be a measurable space. A function P : Σ → [0,1]
satysfying the following conditions
P(Ω ) = 1
A1, A 2 , A n ,... ∈ Σ and A i ∩ A j = ∅, i ≠ j , then
∞  ∞
P U Ai  = ∑ P(Ai )


 i =1  i =1
is called the probability measure on the measurable space Ω, Σ .
The triple Ω, Σ, P
is called the probability space.
Consider B R the smallest σ-field of sets in R that includes all
intervals. This class of sets is called the family of Borel sets in R.
Example. The Lebesgue measure.
Let Ω = [0,1] and B[0,1] be the the family of Borel sets in [0,1].
Let us define the function µ: B[0,1]→ [0,1] as follows
µ([a,b]) = b-a, for 0 ≤ a < b ≤ 1. Function µ can be extended to
all members of family B[0,1]. The function µ is called the
Lebesgue measure in the interval [0,1].
Lebesgue measure is a probability measure: µ([0,1]) = 1.
Similarly the Lebesgue measure can be define in the space R,
but it is no longer a probability measure.
Definition 3.
Let F be a function F: R → [0, ∞). The function F is called a
distribution function (cumulative distribution function) if
a) F is non-decreasing function: a < b ⇒ F(a) ≤ F(b)
b) F is continuous from the left: lim F ( x) = F (a)
c)
lim F ( x) = 0 ,
x → −∞
lim F ( x) = 1
x →a −
x → +∞
Remark. Any distribution function F generates a probability
measure on intervals:
P((a, b]) = F (b) −
lim
x →a +
F ( x)
Conversely. Any probability measure P on
a unique way a distribution function, namely
defines in
F p ( x) = P ((−∞, x))
Definition 4.
A probabilistic measure P on B R is called to be absolutely
continuous with respect to the Lebesgue measure µ if and only if
µ(A) = 0 ⇒ P(A) = 0 for any set A ∈ B R .
The Radon-Nikodym theorem
A probability measure P on B R is absolutely continuous if and
only if there exist a function f P : R → R such that
P( A) =
∫ f P ( x)dx
A
Function f P is called the
density of the measure P
Classification of probability measures
A point x ∈ R is called a discrete mass point of the measure P
if P({x}) > 0. Let I denote the set of all discrete mass points of
a probability measure P. If P(I) = 1, then the measure P is
called the discrete measure.
A measure having no any discrete mass points is called a
continuous measure.
Remark. A probability measure P is continuous if and only if
the distribution function FP of the measure P is a continuous
function.
Theorem (Jordan’s decomposition)
Let P be a probability measure and FP be the distribution
function of P. Then
Fp = dFPd + cFPc , where d ≥ 0, c ≥ 0 and d + c = 1.
The functions FPd and FPc are distribution functions of a discrete
probability measure and continuous probability measures,
respectively.
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