随机变量
概率空间
(
Ω
,
A
,
P
)
(\Omega, {\mathcal{A}}, P)
( Ω , A , P )
X
:
Ω
→
R
X: \Omega \to \reals
X : Ω → R
Ω
\Omega
Ω
{
ω
,
X
(
ω
)
≤
x
}
?
A
\{\omega , X(\omega) \le x\} \subset \mathcal{A}
{ ω , X ( ω ) ≤ x } ? A
Example 1 ：抛一枚硬币10次，令
X
(
ω
)
X(\omega)
X ( ω )
ω
\omega
ω
Example 2 ：
Ω
=
{
(
x
,
y
)
:
x
2
+
y
2
≤
1
}
\Omega = \{(x, y): x^2 + y^2 \le 1 \}
Ω = { ( x , y ) : x 2 + y 2 ≤ 1 }
ω
=
(
x
,
y
)
∈
Ω
\omega = (x, y) \in \Omega
ω = ( x , y ) ∈ Ω
X
(
ω
)
=
x
+
y
X(\omega) = x + y
X ( ω ) = x + y
分布函数
定义 ：对于概率分布
P
P
P
F
F
F
R
→
[
0
,
1
]
\reals \to [0, 1]
R → [ 0 , 1 ]
F是非递减的，
x
1
<
x
2
?
F
(
x
1
)
<
F
(
x
2
)
x_1 < x_2 \rArr F(x_1) < F(x_2)
x 1 ? < x 2 ? ? F ( x 1 ? ) < F ( x 2 ? )
F
F
F
lim
?
x
→
?
∞
F
(
x
)
=
0
,
lim
?
x
→
∞
F
(
x
)
=
1
\lim\limits_{x \to -\infin}F(x) = 0, \lim\limits_{x \to \infin}F(x) = 1
x → ? ∞ lim ? F ( x ) = 0 , x → ∞ lim ? F ( x ) = 1
F
F
F
F
(
x
)
=
F
(
x
+
)
F(x) = F(x^+)
F ( x ) = F ( x + )
离散变量
X
X
X
{
x
i
}
1
∞
\{x_i\}_{1}^\infin
{ x i ? } 1 ∞ ?
f
X
(
x
)
=
P
r
(
X
=
x
)
,
∑
x
∈
X
f
X
(
x
)
=
1
f_{X}(x) = P_{r}(X = x), \sum_{x \in X}f_{X}(x) = 1
f X ? ( x ) = P r ? ( X = x ) , ∑ x ∈ X ? f X ? ( x ) = 1
X
X
X
f
X
(
x
)
f_{X}(x)
f X ? ( x )
x
x
x
f
X
(
x
)
>
0
,
∫
?
∞
∞
f
X
(
x
)
d
x
=
1
f_{X}(x)>0, \int_{-\infin}^\infin f_{X}(x)dx = 1
f X ? ( x ) > 0 , ∫ ? ∞ ∞ ? f X ? ( x ) d x = 1
引理 ：令
F
F
F
X
X
X
P
r
(
X
=
x
)
=
F
X
(
x
)
?
F
X
(
x
?
)
P_{r}(X=x) = F_X(x) - F_X(x^-)
P r ? ( X = x ) = F X ? ( x ) ? F X ? ( x ? )
P
r
(
x
<
X
≤
y
)
=
F
X
(
y
)
?
F
X
(
x
)
P_{r}(x < X \le y) = F_X(y) - F_X(x)
P r ? ( x < X ≤ y ) = F X ? ( y ) ? F X ? ( x )
P
r
(
X
>
x
)
=
1
?
F
X
(
x
)
P_{r}(X > x) = 1 - F_X(x)
P r ? ( X > x ) = 1 ? F X ? ( x ) 如果是连续的，有
F
(
b
)
?
F
(
a
)
=
P
r
(
a
<
X
<
b
)
=
P
r
(
a
≤
X
<
b
)
=
P
r
(
a
<
X
≤
b
)
F(b) - F(a) = P_{r}(a < X < b) = P_{r}(a \le X < b)= P_{r}(a < X \le b)
F ( b ) ? F ( a ) = P r ? ( a < X < b ) = P r ? ( a ≤ X < b ) = P r ? ( a < X ≤ b )
定义 ：有随机变量
X
X
X
F
F
F
F
?
1
(
q
)
=
i
n
f
{
x
：
F
(
x
)
>
q
}
,
q
∈
[
0
,
1
]
F^{-1}(q) = inf\{x：F(x) > q\}, q \in [0, 1]
F ? 1 ( q ) = i n f { x ： F ( x ) > q } , q ∈ [ 0 , 1 ]