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```APPENDIX C
Distribution Functions
Let (Ω, F, P) be a probability space and let X be a random variable on (Ω, F, P).
Theorem C.1. Each random variable X on (Ω, F, P) induces a probability space
(R1 , B 1 , μX ) by
μX (B) = P({ω : X(ω) ∈ B}) = P(X ∈ B)
for all B ∈ B 1 .
(C.1)
dd (C.1) dd μ dd probability measure dddddddd. dddddddd
ddddddd probability space (Ω, F, P) ddddddddddddd probability
space (R1 , B 1 , μX ). ddddddddddddd μ dddddd.
Notation C.2. {ω : X(ω) ∈ B} = {X ∈ B} = X −1 (B). ddd pre-image ddd.
ddddddddd,
μX (B) = P(X ∈ B) = P(X −1 (B)) = (P ◦ X −1 )(B)
i.e., μX = P ◦ X −1
Deﬁnition C.3. The function F deﬁned by
F (x) = μX ((−∞, x]) = P(X ≤ x)
is called the cumulative distribution function (c.d.f.) of the random variable X.
Example C.4.
(1) If the cumulative distribution function of X is given by
x
1
(z − μ)2
F (x) = √
dz,
exp −
2σ 2
2πσ −∞
265
266
C. DISTRIBUTION FUNCTIONS
we say X is normally distributed with mean μ and variance σ 2 , denoted by
X ∼ N (μ, σ 2 ).
(2) Suppose (Ω, F, P) = ([0, 1], B1 , m).
(i) X1 (ω) = ω, for all ω ∈ [0, 1]. Then
F (x) = P(X1 ≤ x) = m({ω ∈ [0, 1] : ω ≤ x}) =
⎧
⎪
⎪
⎪0
⎪
⎪
⎪
⎨
x
⎪
⎪
⎪
⎪
⎪
⎪
⎩1
if x < 0,
if 0 ≤ x ≤ 1,
if x > 1.
(ii) X2 (ω) = 1 − ω, for all ω ∈ [0, 1]. Then
⎧
⎪
⎪
⎪
0 if x < 0,
⎪
⎪
⎪
⎨
F (x) = x if 0 ≤ x ≤ 1,
⎪
⎪
⎪
⎪
⎪
⎪
⎩1 if x > 1,
which is the same as the distribution function of X1 . This random variable
is called uniformly distribution on [0, 1].
Proposition C.5. Let F (x) be the distribution function of a random variable X. Then
(1) F (−∞) := lim F (x) = 0,
x−→−∞
F (∞) := lim F (x) = 1.
x−→∞
(2) F is non-decreasing and right-continuous.
Remark C.6.
(1) If F is diﬀerentiable, the function f (x) = F (x) is called the
probability density function (p.d.f, dddddd) of X.
(2) If F is represented in the form
F =
∞
j=1
bj I[aj ,∞) ,
C. DISTRIBUTION FUNCTIONS
where (an ) ⊆ R, bj > 0, for all j and
∞
267
bj = 1, then F is called a discrete distribution
j=1
function.
Moreover, if ai = aj for all i = j, then the function m({ai }) = bi is called the
probability mass function.
(3) For any distribution function F , we have the following decomposition
F = αFd + βFac + γFsc ,
where α, β, γ ≥ 0, α + β + γ = 1, and
• Fd : discrete distribution function,
• Fac : absolutely continuous distribution function, i.e., Fac (x) exists for all x.
• Fsc : singular continuous distribution function. This means that Fsc exist
almost everywhere, and Fsc (x) = 0 almost everywhere, but
Fsc (x) =
x
−∞
Fsc (y) dy.
For example, the Cantor function shown in Figure C.1 is a singular continuous distribution function.
1
3/4
1/2
1/4
1/9
2/9
1/3
2/3
7/9
8/9
Figure C.1. Cantor function
1
268
C. DISTRIBUTION FUNCTIONS
Remark C.7. The expectation of a random variable X can be written as
E[X] =
X dP =
X(ω) dP(ω) =
Ω
Ω
=
x dμX =
R
∞
=
−∞
x dμX (x) =
R
R
Ω
X(ω) P(dω)
x μX (dx)
x dFX (x).
Theorem C.8 (Change of variables). If X is a random variable and g : R1 −→ R1 is
a Borel-measurable function, then
E[g(X)] =
g(X) dP =
Ω
R
g(x) dμX (x) =
∞
−∞
g(x) dFX (x).
Question : How about the high dimensional case ?
Deﬁnition C.9. Consider the random vecotr X = (X1 , X2 , · · · , Xd ). This implies
that X : Ω −→ Rd . Deﬁne a probability measure on (Rd , B d ) by
μX (B) = (P ◦ X −1 )(B) = P(X −1 (B)) = P((X1 , X2 , · · · , Xd ) ∈ B)
for B ∈ B d .
The distribution function of X is deﬁned by
FX (x) = P(X1 ≤ x1 , X2 ≤ x2 , · · · , Xd ≤ xd )
for x = (x1 , x2 , · · · , xd ).
Theorem C.10. If X is a random variable and g : Rd −→ R is a Borel-measurable
function, then
E[g(X)] =
g(X) dP =
Ω
∞
∞
=
−∞
−∞
···
Rd
∞
−∞
g(x) dμX (x)
g(x1 , x2 , · · · , xd ) dFX (x1 , x2 , · · · , xd )
Theorem C.11. The following statements are equivalent.
C. DISTRIBUTION FUNCTIONS
(1) X1 , X2 , · · · , Xd are independent, i.e.,
P(X1 ∈ B1 , X2 ∈ B2 , · · · , Xd ∈ Bd ) = P(X1 ∈ B1 )P(X2 ∈ B2 ) · · · P(Xd ∈ Bd ).
(2) μX = μX1 × μX2 × · · · × μXd for X = (X1 , X2 , · · · , Xd ).
(3) FX (x1 , x2 , · · · , xd ) = FX1 (x1 )FX2 (x2 ) · · · FXd (xd ).
269
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