• • • • • • • • • •
= f (x, θ ) + f
x θ
2, 3, 5, 7, 11, . . .
•
•
10 10
•
• • • • •
112
15000
{x , x , . . . , x } 1
x , x , 1
2
...,
x
r
r
≤n
xi
f i X
x1 f 1
x2 f 2
2
... ...
i = 1, 2, . . . , r
n
xr fr
1
X
100
1887
− 1985
[0, [0, 1) 14
[a0 , a1 ) f 1
[1, 2) 17
[a1 , a2 ) f 2
[2, 3) 21
... ...
[3, 4) 18
[ar−1 , ar ) fr
[4, 5) 16
[5, 6) 14
,
.
[a0 , a1 ) [a1 , a2 )
f 1 f 2
x1 x2
[ar−1 , ar )
f r
xr
x1 f 1
x2 f 2
• x = a − 2+ a i 1
i
... ...
xr fr
[ai−1 , ai )
i
r
• f
[ai−1 , ai ) i = 1 , 2, . . . , r )
i
5
f i = n
i=1
[18, 25), [25, 35), [35, 45), [45, 55), [55, 65)
x1 t1
x2 t2
... ...
xn tn
,
[18, 25) [18, [25,, 35) [25 [35,, 45) [35 [45,, 55) [45 [55,, 65) [55
xi
ti
95%
0.6
0.4
0.2
0 5
6
7
8
9
10
:
−.
100
OY
20
|
75
OX
75
10 9 8 7 6 5 4 3 2 1 0
2 1 3 1 0 1 4 8
6 5 5 2 3 4 9
6 7 2 5 8
8 779 488 9
1
2
X
Y 0.5
X
3 Z
X
1 14
M
3
1895
1857
− 1936
2 m
p
7
×n
Y
[0, 5) [5, 10) [10, 15) [15, 20) [20, 25) [25, 30)
5 13 23 17 10 2
=k
·
,
k=
.
− − − −
5
20
3
[20,, [20
∞) [15,, 20) [15 15
20
15
20 14. 14.5
20. 20.5
[15,, 20) [15
14. 14.5 20. 20.5
6
17 6
[0, 5) [5, 10) [10, 15) [15, 20) [20, 30)
[18, 25) [25, 35) [35, 45) [45, 55) [55, 65)
5 13 23 17 12
17.5 24.5 34.5 44.5 54.5
− 25.5 − 35.5 − 45.5 − 55.5 − 65.5
8 11 11 11 11
34 76 124 87 64
4.25 6.91 11.27 7.91 5.82
X pie( pie(X )
pie3( pie3(x x)
Nota 5 Nota 6
10% 16%
Nota 7 Nota 8 Nota 9
11%
Nota 10
22% 16%
26%
(6, (6, 6)
Ω Ω
ω Ac Ac = Ω A
A, B, C, . . .
A
\
A
{(i, j); i, j = 1, 6}
(1, 1), 1), (2, (2, 2), 2), (3, (3, 3), 3), (4, (4, 4), 4), (5, (5, 5), 5), (6, (6, 6)} {(1, 6 1
P =
36
=
6
[0, [0, 1]
1749
− 1827
[0, 1] × [0,
36 6
A
6
N N
A
ν N N (A) N
N
f N N (A)
≤ f N N (A) ≤ 1;
(a)
0
(b)
f N N (Ω) = 1;
(c)
c f N N (A ) = 1
(d)
f N N (A
lim f N N (A)
P ( P (A)
ν N N (A)
− f N N (A), ∀A;
B ) = f N N (A) + f N N (B ),
A
B = ∅. A
N
→∞
1929
Ω
∅
∈ F
F ∈ F ∈ F
A A, B
Ac
∈ F ; A
B
Ω
(Ac = Ω A)
\
∈ F
( (
) )
n
(Ai )i=1, =1, n
σ
(An )n∈N
−
σ
∈ F ,
∞
n=1
∈ F ,
i=1
Ai
∈ F .
−
An
F ∈ F ;
F = {A; A ⊂ R} F = {Ω, ∅} A ∈ Ω F = {A, Ac , Ω, ∅} A⊂R F Ω σ− Ω=R
σ
1903
− 1987
Ω
(
σ
)
−
A
P (A) Ω
σ (Ω, (Ω,
F )
σ
F
−
Ω
F
F
σ( )
F
σ( ) =
E
−
E (Rd )
Rd
B
.
A⊃F
B(E )
∈ Bd
σ σ
Bd
F
A
σ
E = A
σ
σ E Rd
F ) → R
P : (Ω, (Ω,
∈ F
A
P ( P (A)
≥ 0, ∀A ∈ F ;
(a)
P ( P (A)
(b)
P (Ω) P (Ω) = 1;
(c)
P ( P (A
∀
∈ F , A
B ) = P ( P (A) + P ( P (B ), A, B
B = ∅,
(Ω, (Ω, P
(Ω, (Ω, (c)
(c)
(An )n∈N
∈ F
Ai P ( P (
An ) =
n
P
σ
P ( P (An ).
∀
A j = ∅, i = j (σ
∈N
F = P (Ω) (Ω) P ( P (A) =
∈N
F )
A
An )
∈ F
(Ω, (Ω,
F , P ) P )
∈Ω
A Ω
F
P (Ω, (Ω,
F , P ) P )
P ( P (A) = 0
P (Ω) P (Ω) = 1
P 1871
F , P ) P )
) (Ω, (Ω,
n
P ( P (
n
−
−
Ω
(b) (Ω, (Ω, F ) F )
n
∈N
A
F )
− 1956
(Ω, (Ω,
F , P ) P )
Ω σ F ⊂ P Ω σ1 Ω ∈ F σ2 A ∈ F =⇒ Ac ∈ F σ3 ∀(An )n∈ ∈ F =⇒ An ∈ F n∈ P : F → R P 1 P (Ω) P (Ω) = 1 P 2 ∀A ∈ F P ( P (A) ≥ 0 P 3 ∀(An )n∈ , An Am = ∅, ∀n =m
N
N
N
∀A ∈ F
F A ⊂ F A − σ
P ( P (A) P ( P (A) = 0 P ( P (A) = 1
→∞
lim lim inf inf An
→∞
An ) =
∈Ω
P ( P (An )
n
∈N
ω
∈N
F
σ
A P A
Ω
lim inf inf An = n
n
A A
(An )n∈N
n
P ( P (
∞
limsup An =
Am
n
→∞
n=1 m n
≥
∞
Am .
n=1 m n
≥
An ⊆ limsup n→∞
(An )n∈N lim An = liminf An = limsup An .
n
→∞
n
→∞
n
→∞
lim lim inf inf An n
An A1 , A2 , . . .
→∞
limsup An n
→∞
[a, b] [a, b] [a, b]
[a, b] [a, b]
[a, b]
[c, d] [c, d]
([c, ([c, d]) d = ([a, ([a, b]) b
P ( P (A) = x
⊂ [a, b] − c. −a
∈ (c, d) ∅
D D D
D ⊂ D
(Ω, (Ω,
F , P ) P )
A, B B
∈ F
P ( P (B ) > 0 P ( P (A B ) P B (A)
A
|
P ( P (A B ) P B (A) = . P ( P (B )
F
P B (A)
(Ω, (Ω,
(Bi )i∈I , (I
∀ ∈ I
P ( P (Bi ) > 0, i
P ( P (A) =
·
∈
⊂ N)
∀A ∈ F .
P ( P (Bi ) P Bi (A),
i I
F , P B )
P ( P (A) > 0 P A (Bi ) =
·
P ( P (Bi ) P Bi (A)
j I
∈
B1 , B2 , . . . , Bn P ( P (B1
∈ F
···
1702
B2
− 1761
·
P ( P (B j ) P BJ (A)
P ( P (B1
··· B2
,
∀i ∈ I.
Bn ) > 0
Bn ) = P ( P (B1 ) P B1 (B2 ) . . . P B1 ··· Bn−1 (Bn ).
·
·
·
Ω
X, Y, Z
ξ,η, ζ
F
E
(Ω, (Ω, , P ) P ) X : (Ω, (Ω, , P ) P )
(E, )
F → (E, E )
B
F−
X
• • •
∈ E , X −1(B) ∈ F
E ≡ (R, B(R)) X (E, E ) ≡ (Rd , B(Rd )) X (E, E ) ≡ (Rn×m , B(Rn×m ) X {(−∞, x], x ∈ R} B(R) X : (Ω, (Ω, F , P ) P ) → R ∀x ∈ R, {ω ∈ Ω | X (ω) ≤ x} ∈ F . = {ω ∈ Ω | X (ω ) ≤ x} {X ≤ x} not {X ∈ B} not = {ω ∈ Ω | X (ω ) ∈ B } X : (Ω, (Ω, F , P ) P ) → Rd F (X ) = {X −1(B), B ∈ Bd} σ− σ− X σ (X ) X F (X n )n∈ X k (ω ) → X (ω ), (E, )
N
F → ∈
E ∈ I )
X i : (Ω, (Ω, , P ) P ) (E, ), (i σ (X i , i I )
σ
− X i , i ∈ I R
X X (ω ) =
i J
∈
xi χAi (ω ),
∀ω ∈ Ω, J ⊂ N.
ω
∈Ω
X
{X i, i ∈ N}
σ
−
−σ −
χA
Ak = X −1 ( xk )
{ }
A n
Ai = Ω
Ai
i=1
X :
∀
A j = ∅, i = j.
xi pi
,
n
pi = P ( P (X = xi ), i
∈ J ⊂ N,
pi = 1
i=1
1 2 3 4 5 6 1/6 1/6 1/6 1/6 1/6 1/6
X
f : (a)
f ( f (x)
(b)
Rd
,
→R
≥ 0,
f ( f (x) dx = 1
R
(c)
P X (B ) =
f ( f (x) dx,
B
f
∀B ∈ F .
X
F → (Rd, B(Rd))
X : (Ω, (Ω, , P ) P )
Bd
X P X (B ) = P ( P (X
P X (B ) =
∈ B), ∀B ∈ Bd. P ( P (A j )δxj (B ),
j J
∈
δa (B ) =
∈B
1, 0,
a
f ( f (x) dx,
∀B ∈ B d ,
X P X (B ) =
B
P X :
Bd → [0, [0, 1]
f ( f (x)
X
X
F : R
F ( F (x) = P ( P (X F ( F (x) = P X ((
−∞, x])
X = (X 1 , X 2 , . . . , Xd ) : (Ω, (Ω, d F : R [0, [0, 1]
→
F ( F (x)
lim F ( F (x) = 0 →−∞
≤ x). (−∞, x]
F , P ) P ) → Rd
F (( F ((x x1 , x2 , . . . , xd )) = P ( P (X 1
• • •
→ [0, [0, 1]
≤ x1; X 2 ≤ x2; . . . , Xd ≤ xd).
≤ F ( F (y), ∀x, y ∈ R, x ≤ y lim F ( F (y) = F ( F (x), ∀x ∈ R y x
lim F ( F (x) = 1.
x
x
→∞
x F ( F (x) =
pi .
{i; xi ≤x} X
f x
F ( F (x) =
f ( f (t) dt,
−∞
∈
x R x) = 1
∈ R. F c : R
F (x), ∀x ∈ R − F (
X φX (t) = φX (t) =
ei t xk pk ,
φX : R X =
k J
∈
ei t x f ( f (x) dx, (i2 =
k J
∈
R
i
x
−1)
X =
[0, 1] → [0,
P ( P (X > x) F ( F (x) = P ( P (X >
→C
xk χAk , (X =
)
• |φX (t)| = 1, ∀t ∈ R • φa X (t) = φX (a t), ∀t ∈ R, a ∈ R • φa X+b(t) = φX (a t)eibt, ∀t ∈ R, a ∈ R • φX (−t) = φX (t), ∀t ∈ R • φX : R → C n
• ∀ti, t j ∈ R, ∀zi, z j ∈ C
− t j )ziz¯ j ≥ 0.
φX (ti
i, j =1
X
X (ω ) =
∀ω ∈ Ω, Ai ∈ F , J ⊂ N X f : R → R pi = P ( P (Ai ), i ∈ J.
xi χAi (ω ),
i J
∈
f ( f (xi ) = pi ,
f ( f (xi ) n
≥ 0, ∀i ∈ J, f ( f (xi ) = 1.
i=1
X
X (ω ) =
i J
∈
E(X )
=
xi χAi (ω ),
∀ω ∈ Ω, J ⊂ N
xi P ( P (Ai ).
i J
∈
X
E(X )
f : R
=
R
xf ( xf (x)dx,
(
).
→R
n
X
X (ω ) =
xi χAi (ω )
X
i=1
E(X )
E(X )
not
=
n
X (ω ) dP ( dP (ω) =
Ω
X : Ω
→R
xi P ( P (Ai ).
i=1
→ R, (n ∈ N) 0 ≤ X 1 (ω ) ≤ · · · ≤ X n (ω ) ≤ X (ω ), ∀ω ∈ Ω X
≥0
X n : Ω
lim X n (ω ) = X (ω ).
n
→∞
E(X )
X : Ω
→R
X =
X + (ω ) = max X (ω), 0 ,
{
= lim lim
→∞ E(X n). − X −
n + X
{−X (ω), 0} = (−X )+(ω).
X − (ω ) = max
}
X E(X ) E(X + )
= E(X + )
−)
E(X
− E(X −),
X X = X 1 + iX 2 : Ω
→C
X E(X )
= E(X 1 ) + iE(X 2 ),
X = (X 1 , X 2 , . . . , Xd )T : Ω
X
E(X )
X : Ω
→ Rd
X
= (E(X 1 ), E(X 2 ), . . . , E(X d ))T .
→ Rd
g : Rd
f )) E(g (X ))
=
g (x)f ( f (x) dx.
Rd
g:R
→R E(X )
=
X (ω ) dP ( dP (ω ) =
Ω
R
X = X X
− E(X )
xf ( xf (x) dx,
R
Ω
X E(X ) = 0
− E(X )
X X
→R
X
X (ω ) =
∈
D2 (X ) =
(xi
i J
∈
X : Ω =m
∈R
xi χAi (ω ),
i J
X
E(X )
→
− m)2 pi,
∀ω ∈ Ω, J ⊂ N, ∀ ∈ J.
pi = P ( P (Ai ), i
R
X
X
2
D (X ) = E[(X [(X
2
− m) ] =
D2 (X )
2
σ =
(X (ω )
Ω
(x
R
− m)2f ( f (x) dx.
σ2
dP (ω ). − m)2 dP ( σ=
√
σ2
E(X )
=m
U (a, b)
•
[0, [0, 1]
X
B(1, (1, 0.5) • (10, 0.5) B(10,
• • •
(m, n) a + (b (b
10 m
×n
U (0, (0, 1)
− a) ∗
[a, b] 35
0
125
232
−1
(a, b) (a, b)
−
[ 2, 3]
N (µ, σ)
• • •
N (0, (0, 1) (m, n)
m
0
•
m+σ
∗
×n
232
N (0, (0, 1)
−1 N (m, σ) N (5, (5, 1.1)
250
200
150
100
50
0 0
2
4
6
8
10
m lege
n
lege
100
× 10
(5, 0.2) N (5,
200
× 50
P (0. (0.01)
10000 µ=2
σ = 0.5
L
• X
B(1, (1, 0.5) • B(30, (30, 0.5)
30
•
p
∈ (0, (0, 1)
moneda
zar
1
1
5/6 3/4 a e t a t i l i b 0.5 a b o r p
a e t a t i l i b 0.5 a b o r p
1/4 1/6
0
1
10
2
10
3
10
4
0
5
10
10
1
10
aruncari
2
10
3
10
4
5
10
10
aruncari
(a) p
3 0.5, 0.2
a, b
c
0.3 X :
x
0.5
[0, [0, 1]
a b c 0.5 0.2 0.3
x < 0.5
,
a
0.5 < x < 0.7
b
X
c
X
6 1, 2, 3, 4, 5
6 [0, [0, 1]
[0, [0, 1]
6
1 1 2 2 3 3 4 4 5 5 (0, (0, ), ( , ), ( , ), ( , ), ( , ), ( , 1) . 6 6 6 6 6 6 6 6 6 6
3 ( 26 , 36 )
[0, [0, 1]
6 .
(b)
∴ ∴
N (b)
1940
1946
b
I =
f ( f (x) dx.
a
f
≥0
f
f
D = [a, b] × [0, [0, d], d > sup f
A
[a, b]
D
f ( f (x) f ( f (x) f ( f (N )
N
P 1909
− 1984
1915
− 1999
f (N ) f ( . N
I
P =
I
, f (N ) · f ( . N
N
5
I =
2
e−x dx.
−2 106
−
[ 2, 5]
2 f ( f (x) = e−x , x ∈ [0, [0, 1]
× [0, [0, 1]
−
N
[ 2, 5]
I = 1.7675
b
I = (b
h(x) = h(x)
− a) 1 b
f ( f (x)h(x) dx,
a
−a 0
,
x
,
.
∈ [a, b], X
I = (b
− a)E(f ( f (X )). )). I
−a I N b
N
f ( f (X k ),
k=1
U [a, b]
X k
V
f ( f (x) dx,
V
⊂ Rn.
∼ U [a, b]
× [0, [0, 1]
2
e−x
2 g (x) = e−x
106
U (−2, 5)
≈ √
1
I =
ex dx
0
e
e = I + I + 1
≈ √
π
N ν N N
π A N
P ( P (A) =
= π4 . π
4 ν N N N π
π
A P ( P (A)
ν N N lim n→∞ N π (
N
1). 1).
√
X
X (ω ) =
∀ω ∈ Ω, J ⊂ N
xi χAi (ω ),
i J
E(X )
=m
∈
∈ J
pi = P ( P (Ai ), i
| | | | − | | − |
αk (X ) = E(X k ) =
xki pi
k );
i J
βk (X ) = E( X k ) =
∈
xi k pi
k);
i J
∈
m)k ) =
µk (X ) = E((X ((X
(xi
− m)k pi
k );
xi
− m|k pi
k);
i J
∈
m k) =
γ k (X ) = E( X
i J
∈
X
m = E(X ) <
| | || | | − − | − | | − | | k
xk f ( f (x) dx =
αk (X ) = E(X ) =
x k f ( f (x) dx =
X k dP
m)k ) =
(x
m)k f ( f (x) dx =
k);
m k) =
x
m k f ( f (x) dx =
X
Ω
R
X
(X
Ω
R
γ k (X ) = E( X
k);
Ω
R
µk (X ) = E((X ((X
X k dP
Ω
R
βk (X ) = E( X k ) =
∞
− m)k dP
);
− m|k dP
);
F ( F (x) α
∈ (0, (0, 1)
α F ( F (xα ) = P ( P (X
xα
≤ xα) = α. xα
α
α
X α
F ( F (x) = α α = 1/2 α = k/100 k/100,, k = 1, 100
j/10 j/ 10,, i = 1, 10
x∗
α = i/4 i/4, i = 1, 4 α = l/1000 l/1000,, l = 1, 1000
f ( f (x∗ )
≤ cr (βr (X ) + βr (Y )) Y )) cr = 1 r ∈ (0, (0, 1] cr = 2r−1 (βr (X )) ))1/r ≤ (βs (Y )) Y ))1/s , ∀0 ≤ r ≤ s; (Lyapunov ) (H lder ) E|XY | ≤ (E|X |r )1/r (E|Y |s )1/s , ∀r, s > 1, r−1 + s−1 = 1; (E|X + X + Y |r )1/r ≤ (E|X |r )1/r + (E|Y |r )1/r ; (Minkowski ) g:R→R g (E(X )) )) ≤ E(g (X )). )). (Jensen ) a > 0, p ∈ N∗ , β p (X ) P ( P ({|X | ≥ a}) ≤ ; (Markov ) a p βr (X + X + Y ) Y )
1857 1859 1864
− 1937 − 1909 1856
− 1918 1859
− 1922
− 1925
r>1
α =
p = 2
X
(X σ2 . a2
{| − m| ≥ a}) ≤
P ( P ( X
(
∈N P ( P ({|X − m| ≥ kσ }) ≤
a = kσ
− m)
(m = E(X )) ))
)
k
1 , k2
{| − m| < kσ }) ≥ 1 − k12 .
P ( P ( X k =3
3σ
{| − m| ≥ 3σ}) ≤ 19 ≈ 0.1.
P ( P ( X
{ − 3σ < X < m + 3σ 3 σ }) ≥
P ( P ( m
8 , 9
X
X
[m
− 3σ, m + 3σ 3σ ]
σ2
m X =
X
−m σ
E(X )
= 0,
D2 (X ) = 1.
X X X, Y X + X + Y
mX , mY D2 (X + X + Y ) Y ) = = =
Y 2 σX
2 σY
− (mX + mY )2)] [(X − mX )2 ] + E[(Y [(Y − mY )2 ] + 2E[(X [(X − mX )(Y )(Y − mY )] E[(X D2 (X ) + D2 (Y ) Y ) + 2 E[(X [(X − mX )(Y )(Y − mY )]. )]. [(X + + Y E[(X
1821
− 1894
Y
X (X, Y ) Y ) = E[(X [(X
Y
(X, Y ) Y )
− mX )(Y )(Y − mY )] D2 (X + X + Y ) Y ) = D2 (X ) + D2 (Y ) Y ) + 2
(X, Y ) Y ).
−
(b) (X, Y ) Y ) = (Y, X ) = E[(X [(X mX )(Y )(Y 2 (c) (X, X ) = D (X ) X (d) (X + X + Y, Z ) = (X, Z ) + (Y, Z ) (e) X Y (X, Y ) Y ) = 0
Y X
2 σX
Y X =
X
− mX
Y =
σX
Y
− mY
− mY )] = E(X Y ) Y ) − mX mY X,Y,Z X
2 σY
σY
X X
Y ρ(X, Y ) Y ) = X
Y
(X, Y ) Y ) . σX σY
(X , Y ) Y ) =
Y
− ρ(X, Y ) Y ) = 0. (b) (c)
− 1 ≤ ρ(X, Y ) Y ) ≤ 1,
X
Y = aX + b (a, b ρ(X, Y ) Y ) =
(Ω, (Ω, A
∈ R)
F , P ) P ) A
Y
+1, +1, 1,
−
a = 1; a = 1.
−
A, B B
∈ F
B
A=
B= B B
P ( P (B ) > 0 A
A B
|
P ( P (A B ) = P ( P (A),
P ( P (A B ) = P ( P (A). P ( P (B )
P ( P (A
A, B
·
B ) = P ( P (A) P ( P (B ). P ( P (B )
∈ F
{i1, i2, . . . , ik } P ( P (Ai1
A1 , A2 , . . . , An 1, 2, . . . , n
{ ···
}
Ai1
·
·
·
Aik ) = P ( P (Ai1 ) P ( P (Ai2 ) . . . P ( P (Aik ).
A1 , A2 , . . . , An
Ai
A j
· ⊂ F ⊂
P ( P (Ai (Ai )i∈I
A j ) = P ( P (Ai ) P ( P (A j )). )).
, (I
N),
P ( P (
A j ) =
j J
j J
∈
J
P ( P (A j ),
∈
⊂ I , J − A
B
C A B
P ( P (A
1 C ) = P ( P (A) P ( P (C ) = ; 4
·
P ( P (B
0 = P ( P (A
1 C ) = P ( P (B ) P ( P (C ) = ; 4
B
·
·
1 B ) = P ( P (A) P ( P (B ) = . 4
·
Mi ⊂ F
j J
∈
1 C ) = P ( P (A) P ( P (B ) P ( P (C ) = . 8
{Mi, i ∈ I ⊂ N} A j ∈ M j P ( P (
P ( P (A
A j ) =
j J
∈
−
σ J
P ( P (A j ).
C
⊂ I
·
σ
F → R,
(X i )i∈I : (Ω, (Ω, ) X i σ (X i ) i∈I (X i )i∈I : (Ω, (Ω, ) R, (I N),
−
{ } F →
(I
⊂
⊂ N), σ−
≥2 B1 , B2 , . . . , Bk ∈ Bd , {X 1 ∈ B1} {X 2 ∈ B2} · · · {X k ∈ Bk } = P ( P (X 1 ∈ B1 ) · P ( P (X 2 ∈ B2 ) · . . . · P ( P (X k ∈ Bk ), {X 1 ∈ B1}, {X 2 ∈ B2}, . . . , {X k ∈ Bk } k
P
X 1 X 2
{1, 2, 3, 4, 5, 6} X i : Ω
P
{
X 1 = i
} {
X 2 = j
}
→ {1, 2, 3, 4, 5, 6},
i = 1, 2.
{ } 361 P ( P ({X 1 = i}) · P ( P ({X 2 = j }), ∀i, j ∈ {1, 2, 3, 4, 5, 6},
= P ( P ( X 1 = i, X 2 = j ) = =
X 1
X 2
{X 1, X 2, . . . , Xn }, (i) (ii) ii) (iii) iii) (iv) iv)
X 1 , X 2 , . . . , Xn
F → R, i = 1, n
X i : (Ω, (Ω, )
;
∈ B1, X 2 ∈ B2, . . . , Xn ∈ Bn) = P ( P (X 1 ∈ B1 ) · P ( P (X 2 ∈ B2 ) · . . . · P ( P (X n ∈ Bn ), ∀Bi ∈ Bd ; F (X , X ,..., ,..., X ) (x1 , x2 , . . . , xn ) = F X (x1 ) · F X (x2 ) · . . . · F X (xn ), ∀x1 , x2 , . . . , xn ∈ R; n φ(X , X ,..., ,..., X ) (t) = φX (t1 ) · φX (t2 ) · . . . · φX (tn ), ∀t = (t1 , t2 , . . . , tn ) ∈ R .
P ( P (X 1 1
2
n
1
2
n
1
1
2
2
n
n
X 1 , X 2 , . . . , Xn
|X k |) < ∞, ∀k = 1,1 , 2, . . . , n, n,
E(
|X 1 · X 2 · . . . · X n|) < ∞ E(X 1 · X 2 · . . . · X n ) = E(X 1 ) · E(X 2 ) · . . . · E(X n ).
E(
X 1 , X 2 , . . . , Xn D2 (X k ) < D2 (X 1 + X 2 + . . . + X n ) <
∞, ∀k = 1, 2, . . . , n, n,
∞
D2 (X 1 + X 2 + . . . + X n ) = D2 (X 1 ) + D2 (X 2 ) + . . . + D2 (X n ).
(Ω, (Ω, σ2
F , P ) P )
X n , X : Ω
X n
X
X n
→R
m
−a.s.→ X
P ( P ( lim X n = X ) = 1, n
→∞
∃ Ω0 ∈ F , P (Ω P (Ω0 ) = 1, X n
∀ ∈ Ω0 .
lim X n (ω ) = X (ω ), ω
n
X
X n
→∞
prob
−→ X
∀ε > 0, nlim P ({ω : |X n (ω ) − X (ω)| ≥ ε}) = 0. 0. →∞ P ( X n
r
X
lim
n
→∞
X n
|
X n (ω )
Ω
r
−L→ X
dP (ω ) = 0, − X (ω)|r dP (
|
− x|r f ( f (x)dx = 0. n→∞ −rep→ X, X n ⇒ X X lim E(g (X n )) = E(g (X )), )), ∀g : R → R, n→∞ lim
xn
R
X n
X n
X lim F Xn (x) = F X (x),
n
X n
→∞
∀x
F X .
X lim φXn (t) = φX (t),
n
(a) (b)
X n
r
−L→ X (c)
(d)
.
→∞
−a.s.→ X prob X n −→ X prob X n −→ X
X n
∀t ∈ R. X n
prob
−→ X. .
X n
⇒ X.
f ( f (x)
F ( F (x)
F −1 (y )
f ( f (x)
X x
F −1 (y )
F ( F (x)
F ( F (x, θ)
θ
∈R P ( P (X
≤ x) = F ( F (x) θ). P ( P (X < x)
P ( P (X X
P (X < x) + P ( P (X = x) = P ( P (X < x). ≤ x) = P (
∼ N (5, (5, 2) P ( P (X < 4) =
.
X P ( P (X < x) x) =
P ( P (X P ( P (X
≤ [x]) ≤ m − 1)
,x ,x = m
∈ Z,
[x]
x X
∼ B(10, (10, 0.3) P ( P (X < 5) = P ( P (X =
≤ 4)
= 0.8497 8497..
N (µ, σ)
B(n, p) BN ( N (n, p) P (λ) U (n) G eo( eo( p) p) H(n,a,b) n,a,b)
U (a, b)
exp( exp(λ) Γ(a, Γ(a, λ) β (m, n) log (µ, σ ) 2 χ (n) t(n) (m, n) W bl( bl(k, λ)
N
F
100 (a) (b)
X
52 P (45 P (45
≤ X ≤ 55) 52)) P = P P ((X = 52
(100,, 0.5) B(100
X
52 P = C 100 (0..5)52 (0 (0 (0..5)48 = 0.0735 0735..
·
·
F X
X P (45 P (45
≤ X ≤ 55)
≤ 55) − P P ((X < 45) F X (55) − F X (44) 55 k · (0 C 100 (0..5)k · (0 (0..5)100−k = 0.7287 7287..
= P P ((X = =
k=45
√
20 15 T
F T T exp((λ) λ = 20 ∼ exp P ((T ≥ 15) = 1 − P P P ((T < 15) = 1 − F T (15),, T (15) T
P ((T P
≥ 15)
= 0. 0 .4724 4724,, 47..24% 47
√
20% 7 3 7
X P ((X P
≥
7
∼B
X (7,, 0.2) (7 3) = 1 P P ((X < 3) = 1
−
P ((X = 0) = C 70 p0 (1 P
−
p = 0.2 P ((X 2) = 1 P
≤
− F X (2) = 0.0.1480
− p p))7 = 0.2097
X √
(a)
20 15
(b)
10
200
≥ 15) = 1 − P P ((X < 14) = 1 − F X (14) = 0.0.8951 10 X k < 199) = 1 − F X (199) = 0. 0.5094 5094,, X k ∼ P (200) (200)
P 1 = P P ((X 10
P 2 = P P ((
k=1
10
X k
≥ 200) = 1 − P P ((
k=1
k
k=1
3
4
5
8
(Ω, (Ω,
F , P ) P )
X : (Ω, (Ω,
(X n )n∈N : (Ω, (Ω,
F , P ) P ) → R
F , P ) P ) → R X
X 1 , X 2 , . . . , Xn , . . .
∀x ∈ R.
F X1 (x) = F X2 (x) = . . . = F Xn (x) = . . . ,
n
S n =
X k
k=1
(X n )n S n
− E(S n) prob −→ 0, n
S n
(
− E(S n) −a.s.→ 0), 0), n
(n
→ ∞)
(X n )n∈N∗ X n 1 lim 2 D2 (S n ) = 0 n→∞ n S n
2
− E(S n) prob −→ 0,
n
n
β2 (X n ) <
∞
→ ∞. S n n
a>0
P ( P (
S n n
−E
≥ ≤ S n n
a
1 2 D a2
S n n
=
1 1 2 D (S n ) a2 n2
→ 0,
n
→ ∞.
X n
E(X n )
=
∀ ∈N
m, n
S n n
prob
−→ m.
m
A N
ν N N
A
ε> 0 lim P
n
→∞
N
− ν N N N
P ( P (A) = p
p < ε = 1.
N
p
ν N N
1 i X i = X i
X i
1, 0,
i
A
i
;
A
∼ B(1, (1, p) n
X i = ν N N
i=1
∼ B(N, p),
E(ν N N )
D2 (ν N N ) = N p(1
= N p, p,
− p) p).
ν N N N P
P
X n , n
− ≥− − ≥ − −
ν N N N
E
ν N N N
≥1
ν N N N
<ε
p < ε
1
1
D2
ν N N N ε2
,
p(1 p) p) , N ε2
1 (X n )n 1 n
n
X k
k=1
∀ ∈ N∗
m = E(X n ), n
1894
− 1959
prob
−→
m, (n
→ ∞),
(X n )n∈N∗ m = E(X 1 )
|X 1|) < ∞
E(
(X n )n 1 n
n
−a.s→
X k
k=1
m, (n
X 1 + X 2 + P lim n→∞ n
→ ∞).
· · · + X n = m
(X n )n∈N
= 1.
m = E(X 1 )
√
n
1
σ n
X k
k=1
− nm ⇒ Y ∼ N (0, (0, 1), 1),
n
σ 2 = D2 (X 1 )
→ ∞.
n
−√
S n nm σ n
S n =
N (0, (0, 1)
n
1 X = n not
Z n =
√ k=1
σ n
→∞
n
1
lim P ( P (Z n
n
N (m, √σn )
X k
X k
k=1
− nm
.
≤ x) = Θ(x Θ(x), ∀x ∈ R,
Θ(x Θ(x)
≤
lim P a
n
→∞
−√
S n nm σ n
≤ b
=
1 2π
√
b
2 /2
e−x
dx = Θ(b Θ(b)
a
m = 0, σ2 = 1 1 n
n
√ k=1
X k
⇒ Y ∼ N (0, (0, 1), 1),
n
→ ∞.
− Θ(a Θ(a).
n
{X k }k
n
{X k }k
n X k
X 1 , X 2 , . . . , Xn , . . . X 1 + X 2 + + X n
···
B(1, (1, p)
−∞ < a < b < ∞
≤ √− ≤
lim P a
→∞
≥ 30
30
n
n
n
∈ N∗
S n
np npq
b
E(S n )
=
1 2π
√
b
2 /2
e−x
dx.
a
(q = 1
S n =
− p) p)
D2 (S n ) = npq.
= np
n np
npq
• •
np
n
≥5
n(1
− p) p) ≥ 5
p P ( P (X
≤ k) ≈ Θ
√− k
np , npq
Θ Θ(x Θ(x) =
1 2π
√
P ( P (X = k) Φ(x Φ(x) =
x
e−
y2
dy ,
2
x
−∞
∈ R.
√−
1 k np Φ , npq npq
≈√
d Θ(x) dx Θ(x
P ( P (X 1667
− 1754
≤ k) ≈ Θ
1 2
− np √npq
k+
.
1 2
P ( P (X = k) = P ( P (k = P
≈
Θ
− 12 < X < k + 12 ) k − 12 − np k + 12 − np X − np < √ < √ √ npq
npq
1 2
− np − Θ √npq
k+
npq
−− k
1 2
√npq np
. µ
P ( P (X
− ≤ ≈ − ≈ − −− k)
Θ
≈Θ
k+
k+
1 2
1 2
µ
σ
1 Φ σ
P ( P (X = k)
P ( P (X = k )
k
µ
,
σ
µ
k
−Θ
σ
1 2
σ
µ
.
X F X (x)
f X (x)
g (x)
Y = g (X ) DY = x
{ ∈ R; g(x) ≤ y} {Y ≤ y} = {g(X ) ≤ y} = {ω ∈ Ω, X (ω) ∈ DY } (not = {X ∈ DY }). g (X )
F Y P (X Y (y ) = P ( =
∈ DY ),
f X (x) dx.
DY
g (x)
σ2
not
x = h(y) = g −1 (y) Y
dh( dh(y) f Y . Y (y ) = f X (h(y )) dy
g (x) = ax + b, a = 0. f X (x)
X Y = g (X ) 1 f Y f X Y (y ) = a
||
g (X ) Y
F Y Y (y )
P
F Y P (aX +b Y (y ) = P (
≤ y) =
F X
P
−b X ≤ a y−b X ≥ y
a
f Y Y (y ) =
,
− y
b
a
f Y Y (y ) a > 0; a < 0;
dF Y 1 Y (y ) = f X dy a
||
− − − − y
F X
= ,
.
a
1
F X
y
b
a
b
y
b
a
,
a > 0;
,
a < 0;
.
X F ( F (x)
0
U (0, (0, 1) F Y Y
1
U
Y = F −1 (U ) Y
X
F Y Y
X F Y P (Y Y (x) = P ( = P ( P (U
≤ x) = P ( P (F −1 (U ) ≤ x) F (x)) = F ( F (x), ∀x ∈ [0, [0, 1]. 1]. ≤ F (
X F −1 (U n )}
U (0, (0, 1)
{U 1
X
}
U 2 . . . U n F −1 (U 2 ) . . .
{F −1(U 1)
100
• •
X
52 P (45 P (45
≤ X ≤ 55) P = P ( P (X = 52 52))
X
B(100, (100, 0.5) 52 P = C 100 (0. (0.5)52 (0. (0.5)48 = 0.0735 0735..
·
P =
·
1 Φ 100 0.5 0.5
√ · ·
P = Θ
1 2
√
− · ·
52 50 100 0.5 0.5
√25− 50 − Θ
52 +
52
≈
0.0737 0737..
−√12 − 50 ≈ 0.0736 0736.. 25
F X
X P (45 P (45
≤ X ≤ 55)
≤ 55) − P ( P (X < 45) F X (55) − F X (44) 55 k · (0. C 100 (0.5)k · (0. (0.5)100−k = 0.7287 7287..
= P ( P (X = =
k=45
P (45 P (45
≤ X ≤ 55) ≈ Θ
1 2
√25− 50 − Θ
55 +
45
1 2
−√ − 50 25
= 0.7287 7287..
√
O 0.5
X i X = 1
−
0.5 100
∈
i i N X 0.5 100
X = 1 E(X )
=0
n
D2 (X )
S n =
−1
=1
1
X i
i=1 n
E(S n )
=
E(X i )
n
2
=0
D (S n ) =
i=1
{X i}i=1, =1, n
n = 100 S 100 100
|
P ( P ( S 100 100
D2 (X i ) = n,
i=1
n (0, 10) ∼ N (0,
≥ 30
P (−2 ≤ S 100 | ≤ 2) = P ( 100 ≤ 2) ≈ Θ
2+
S n 1 2
− 0 − Θ −2 + 12 − 0 10 10
√ ∼ N (0, (0, n)
= 0.1583 1583..
√
1D
ν n n
P n
ν n n
P
n
ν n n
− 0.5
ν n
∼ B(n, 0.5)
E(
ν n ) = 0.5, n X =
≤ ≥
− 0.5
ν n n
1
−
P n ν n
0.98 = P = P = Θ = Θ Θ
5
D2 (ν n ) =
n 4
ν n 1 )= . n 4n
a = 0.1 1
−
D2 ν nn =1 0.01
− 25n .
− 25n ≥ 0.98, 98,
0.1
≤ ν nn − 0.5 ≤ 0.1
= 0.99
√ n 5
= 0.98. 98.
− E(ν n) = ν n −√0.5n ∼ N (0, (0, 1). 1).
D(ν n )
√ n
n 2
=
≥ 1250
n
0.98. 98.
E(ν n )
D2 (
≤ ≥ 0.1
0.1
0.5 n
− ≤ − ≤ − √ ≤ −√ ≤ √ √ − − √ √ − − √ √ − ν n 0.5 0.1 n n ν n 0.5n 0.1 0.5 0.5 n n n Θ 5 5 n n 1 Θ 5 5 0.1
= Θ(0. Θ(0.99) = z0.99 n
0.1
n 0.5
= 2Θ
≈ 2.33
n 5
1
0.99
≈ 135 135..2974
n
≥ 136 √
25%
0.97 20%
30% ν n n
n
∈N
≤ ν nn ≤ 0.3 ≥ 0.97, 97,
P 0.2 P
ν n n
ν n
P
n
∼ B(n, 0.25)
− 0.25
0.05
1
0.97. 97. E(ν n )
n 4
=
D2 (ν n ) =
n
1
−
D2 ν nn =1 0.052
− 75n .
− 75n ≥ 0.97, 97,
−
0.05
ν n n
≤ − 0.25 ≤ 0.05
= 0.97. 97.
25n − E(ν n) = 4 ν n −√ 0.25n ∼ N (0, (0, 1). 1).
D(ν n )
3n
− ≤ − ≤ − × ≤ −√ ≤ × − − − − − ≈
0.97 = P
0.05
ν n n
= P
0.05
4
n 3 n 3
= Θ 0.2 = Θ 0.2
n 3
3n 16
≥ 2500 P
Θ 0.2
0.05
≤ ≥
ν n n
n
ν n
− 0.25
ν n ν n 3 ) = 0. 0 .25, 25, D2 ( ) = . n n 16n 16n X = ν nn a = 0.05
E(
≤ ≥
= 0.985
0.2
n 3
0.25
n 3
Θ
1
D(ν )
n
n
0.05
0.25n 25n n 0.05 4 3 3n n 0.2 3 n n Θ 0.2 = 2Θ 0.2 3 3 4
D2 (ν )
ν n
= z0.985 2.17 n 353 353..1969
1
0.985
≈
n
≥ 354
√
N F
X
n
x1 , x2 , . . . , xn xi X
∈ R, i = 1, 2 . . . , n, n ≤ N
{x1, x2, . . . , xn} 1 x ¯= n
n
xi ,
i=1
{x1, x2, . . . , xN }
N
1 µ= N
x ¯ i
di = xi
−
N
xi
i=1
µ
{x1, x2, . . . , xN }
µ x n
(xi
i=1
k
∈ N∗
− x) = 0.
k 1 αk = n
n
i=1
xki
.
p = 0 x ¯ p =
1 n
n
x pi
1 p
,
i=1
x ¯
p =
x ¯(q) x ¯(g) =
x¯(h)
−1
x ¯(h)
√x1 · x2 · . . . · xn, n
≤ x¯(g) ≤ x¯ ≤ x¯(q). k
1 mk = N k
N
xki ,
(k
i=1
∈ N∗).
∈ N∗
k 1 µk = n
n
(xi
i=1
− x)k ,
,
N
1 µk = N
(xi
i=1
− µ)k ,
.
{x1, x2, . . . , xn} s2 =
n
1 n
− 1 i=1
(xi
− x¯)2
=
1
n
1 σ = N
1 n
n
(xi
i=1 s2
− x¯)2 σ2
N
i=1
(xi
2
2
− 1 [ i=1 xi − n(x¯) ]
N 2
n
− µ)2.
.
p = 1 p = 2
{x1, x2, . . . , xn} s= N
− − − n
1
n
1
x ¯)2 .
i=1
N
1 N
σ=
(xi
(xi
µ)2 .
i=1
−x
x Z
x
{x1, x2, . . . , xn}
Z x
z=
s
Z
x
z=
n
− x¯ . − µ. σ
(x1 , y1 ), (x2 , y2 ), . . . , (xn , yn ) cov
cov
r
=
n
1
n
)(yi − y¯). − 1 i=1 (xi − x¯)(y
1 = N
N
(xi
i=1
)(yi − µy ). − µx)(y
cov , sx sy
r
=
=
cov , σx σy
, .
X F n∗ : R −→ [0, [0, 1] F n∗ (x) =
{i; xi ≤ x} . n
{x1, x2, . . . , xn}
Ω
X
F ( F (x)
X
X
F n∗ (x) F n∗ (x)
−prob → F ( F (x),
n
{X ≤ x}
A
{x1, x2, . . . , xn}
→ ∞, ∀x ∈ R.
p = P ( P (A)
n
A
ν n = n
{i; xi ≤ x} = F ∗(x). n
n
γ 1 =
µ3 3/2
µ2
.
γ 1 = 0 γ 1 < 0
γ 1 > 0
n
1/2
n γ 1 =
(xi
i=1
n
(
(xi
i=1
− x)3 2 3/2
− x) )
µ3 γ 1 = 3 = σ
N
(xi
i=1
− µ)3
.
K =
µ4 µ22
− 3.
n
− −
n
(xi
x)4
i=1 n
K =
(
(xi
x)2 )2
−3
i=1
µ4 K = 4 σ
K = 0
−
1 3= n σ4
N
(xi
i=1
− µ)4 − 3
,
−
( 3) 0 K<0
K>0
K>0
K<0
n
n
n=2
2 x1 < x2 < x0.5 =
x(n+1)/ +1)/2 (xn/2 n/2 + xn/2+1 n/2+1 )/2
n= 4
x0.5
· · · < xn n= n= 3
x0.25 x0.75
Q3 99
n = 10 n = 1000
− Q1
9
n = 100 999
x∗ 6 [35,, 45) [35
Q1 Q3
n
n
X
{x1, x2, . . . , xn}
n
. . . f n
}
{f 1
f i = n
i=1
1 x ¯f = n s2 =
1
n
n
−
xi f i ,
i=1 n
1
f i (xi
i=1
− x¯f )2 = n − 1 √2
s=
s ,
xi2 f i
i=1
Me = l + l
n 2
− n x¯f 2
,
− F M e c, f M e
n
F M e
f M e
c
M od = l + d1
n
1
d1 c, d1 + d2
d2 l
c
m = 400 n n
{x1, x2, . . . , xn}
n 1 x= n x
n
i=1
xi .
f 2
x
≈m m
x x x x
α x0.75
−x
k
p x x
0.25
x x x x x x x x x x x x y y
n lim C nk pk qn−k =
n p
→∞ →0
e−λ λk . k!
λ=np
0.12
0.1
0.08
0.06
0.04
0.02
0 0
5
10
15
B(n, p) P (np) n
20
25
30
n = 100, p = 0.15
≥ 30, 30, p ≤ 0.1, λ = n p ≤ 0.1.
3σ
X 1 − e−λ x ,
x>0
F −1 F −1 (u) =
{u1
u2 . . . un − − 1 1 F (u2 ) . . . F (un )
}
∼
λ)
−
λ ln(1
0
− u)
F : R
, u ,
.
[0, [0, 1] λ) 150
∼ U − π2 , π2
F ( F (x) =
∈ (0, (0, 1);
}
X
−→ [0, [0, 1], 1],
{F −1(u1) 5)
Y = tan X Y
(x) x
20
0
10
(x)
x x
(x)
(x)
x
±∞
k n 1, 2, . . . , n
{
k
10
50
}
100
X
{1, 2, . . . , n} 20
(x)
−
d = Q3
−
525 50 = 475 Q1 = 84 Q1 = 56
154.15 154. 105..75 105 Q3 = 140 Q2 = M e
100
340..5 475 340 475..5 525
√
a = 30 x ¯=
s2 = =
·
(x f ) f ) 1 = (2. (2.5 70 n
1
12.5 × 23 + 17. 17.5 × 17 + 22. 22.5 × 10 + 27. 27.5 × 2) = 13. 13.9286 9286.. × 5 + 7.7.5 × 13 + 12.
· − n·x ¯2 ) n−1 1 (2. (2.52 × 5 + 7. 7.52 × 13 + 12. 12.52 × 23 + 17. 17.52 × 17 + 22. 22.52 × 10 + 27. 27.52 × 2 − 70 · 13. 13.92862 ) 69 (
(x2 f ) f )
= 37. 37.06. 06.
[10,, 15) [10
[0, [0, 5)
[5, [5, 10)
35
35 17
5 + 13 = 1188
17 23
M e = 10 +
17 23
× 5 = 13. 13.6957 6957..
[10,, 15) [10 70
12. 12.5 18
Q1 = 10
√
52
(a) (b) (c) (d) (e)
0 7 0.1346
1 9 0.1731
52
x=
xi = 2.1731 1731,,
i=1
F n∗ (x) = P ( P (X
s=
2 14 0.2692
1 51
≤ x) =
3 12 0.2308
4 10 0.1923
52
(xi
i=1
0, 7 52 , 16 52 , 30 52 , 42 52 ,
1,
− x¯)2 = 1.1 .3094 3094,, F n∗ (x)
x < 0; x [0, [0, x [1, [1, x [2, [2, x [3, [3, x 4.
∈ ∈ ∈ ∈ ≥
1); 2); 3); 4);
M e = 2.
(e) P ( P (X
16 P (X < 2) = 1 − P ( P (X ≤ 1) = 1 − F n∗ (1) = 1 − = 0.6923 6923.. ≥ 2) = 1 − P ( 52
√
x = 7.24
s = 0.7
8.45 s Z z=
x
− x = 8.50 − 7.24 = 1.8 σ
0.7
. √
χ2
•
• •
=
•
•
1, 3, 5, 7, 9 1 9
v
n
• []
2
•
j
i
•
m n
×
m n
×n
• f ( f (x, y) = e5x sin3y sin3y
f (7 f (7,, π)
•
1
• • • •
X
rand normal 7 5
−10
10
2
[0, [0, 3π ] X
2 X 0.01
X X X X X X ex ln(x ln(x)
n! A
m
×n
1 m
×n
n 3
7 20 n
×n ×3
A A A A A A k 105 e
3 5
X X
t 3
m
×n
&
z
U (n) X
∼ U (n)
X
{1, 2, . . . , n}
P ( P (X = k) = E(X )
=
n+1 2 ,
1 , n
k = 1, 2, . . . , n. n.
n2 1 12 .
−
D2 (X ) =
U (6) (6)
B(1, (1, p) X
(1, p) ∼ B(1,
X = 1 P ( P (X = 1) = p; P ( P (X = 0) = 1 2 p) E(X ) = p; D (X ) = p(1 p)
−
X = 0
− p B(1, (1, 0.5)
B(n, p) X
∼ B (n, p)
(n > 0, p
{0, 1, . . . , n}
X
(0, 1)) ∈ (0,
P ( P (X = k) = C nk pk (1 E(X )
(X k )k=1,n =1,n
∼ B(1, (1, p)
= np; np;
− p) p)n−k , k = 0, 1, . . . , n. n. D2 (X ) = np(1 np(1 − p) p) n
(X k )k
X =
k =1
15
X k
∼ B(n, p) B(15, (15, 0.5)
H(n, a, b)b) X
∼ H(n, a, b)b)
(n, a, b > 0)
C ak C bn−k P ( P (X = k ) = , C an+b 1654
− 1705
k
max(0, max(0, n
min(a, n). − b) ≤ k ≤ min(a,
n
EX =
E(X i )
= np; np; D2 (X ) = np(1 np(1
i=0
(X k )k=0,n =0,n
∼ B(1, (1, n)
p=
a+b−n − p) p) a+b−1
a a+b
n
X =
X i
i=1
∼ H(n, a, b)b). n
D2 (X )
D2 (X i )
(X i )i
i=0
N = a + b
a+b n a+b 1
− −
n D2 (X )
N n N 1
− −
n np(1 − p) p) 1 − ≈ np(1 N
.
N
− n N
1
≈ a+a+b−b n = 1 − N n
N
N
P (λ) λ λ>0
X
∼ P (λ)
P ( P (X = k ) = e−λ E(X )
X λk , k k!
∀ ∈ N.
= λ; D2 (x) = λ eo( p) p) G eo(
p X
∼ G eo( eo( p) p),
( p
∈ (0, (0, 1))
P ( P (X = k) = p(1 E(X )
=
1
− p ;
p
D2 (X ) =
X
− p) p)k ,
N
k
∈ N,
−
1 p p2
X
∼ G eo( eo( p) p)
1781
− 1840
n
Y = X + X + 1
p
≥ 0.
→∞
BN (m, p) m m=1 m
≥ 1, p ∈ (0, (0, 1)
∼ BN (m, p) {m X m−1 m P ( P (X = k ) = C m p)k , ∀k ≥ m, p ≥ 0. +k−1 p (1 − p) m(1 − p) p) m(1 − p) p) E(X ) = ; D2 (X ) = X
m + 1 m + 2, . . .
}
p2
p
U (a, b) X
∼ U (a, b) (a < b f ( f (x; a, b) =
E(X )
=
1 b a
, ,
−
0
x .
∈ (a, b)
a+b (b a)2 , D2 (X ) = 2 12
−
(0, (0, 1)
U (0, (0, 1) N (µ, σ) X
∼ N (µ, σ),
X (x−µ)2 1 − f ( f (x; µ, σ) = e 2σ2 , x σ 2π
√
E(X )
=µ
∈ R.
D2 (X ) = σ 2 µ = 0, σ2 = 1 f ( f (x) =
√12π e−
x2
2
, x
∈ R. N (0, (0, 1)
X σ=1 (µ 3σ, µ + 3σ ) = ( 3, 3)
−
X µ σ
−
∼ N (0, (0, 1) (0, 1) ∼ N (0, Z
−
X = σZ + µ (0, (0, 1) Θ(x Θ(x)
N
∼ N (µ, σ)
Θ(x Θ(x) =
1 2π
√
X
x
−∞
e−
y2
2
dy.
∼ N (µ, σ)
Z =
X
X
∼ N (0(0, σ),
(σ = 1, 2, 3)
∼ N (µ, σ) F ( F (x) = Θ(
x
− µ ), σ
x
∈ R.
N (µ, σ)
log
f ( f (x; µ, σ) = Y
∼ log N (µ, σ)
X
∼ N (µ, σ)
− (ln 2xσ−2µ)2 1 √ e xσ 2π 0
∼ N
ln Y (µ, σ) 2 E(X ) = eµ+σ /2 ,
Y = eX , ,
x>0 x 0
≤
2
D2 (X ) = e2µ+σ (eσ
2
− 1)
exp( exp(λ)
P (λ) X
∼ exp( exp(λ)
(λ > 0) f ( f (x; λ) = E(X )
{
=
1 λ
D2 (X ) =
P ( P ( X > x + y
λe−λx , 0 , 1 λ2
x>0 x 0
≤
}|{X > y }) = P ( P ({X > x }), ∀x, y ≥ 0.
Γ(a, Γ(a, λ) X
∼ Γ(a, Γ(a, λ)
a, λ > 0 f ( f (x; a, λ) =
λa a 1 e λx Γ(a Γ(a) x
x > 0,
0
x
− −
≤ 0.
Γ Γ : (0, (0, E(X )
Γ(1, Γ(1, λ)
=
(0, ∞), ∞) → (0,
Γ(a Γ(a) =
∞
xa−1 e−x dx.
0
a a , D2 (X ) = 2 . λ λ
≡ exp(λ exp(λ)
n
exp(λ) {X k }k=1,n =1,n ∼ exp(λ
X k
k=1
W bl( bl(k, λ)
k=1
k = 3.4 X
∼
→∞
k W bl( bl(k, λ) (k > 0, λ > 0) f ( f (x; k, λ) = X
χ2 X
k λ
0
bl(k, λ) ∼ W bl(
− −( λx )k
x k 1 e λ
E(X )
, ,
≥
x 0 x < 0.
= λΓ 1 +
1 k
χ2 (n)
∼ χ2(n) f ( f (x; n) =
Γ E(χ2 )
1 n Γ( 2 )2 2 n
0
= n, D2 (χ2 ) = 2n. 1887
− 1979
n
x
x 2 −1 e− 2
x > 0, x
≤ 0. n
Γ(n, λ) ∼ Γ(n,
X k
Γ( n2 , 12 ) k = 1, 2, . . . , n
χ2 (n) (0, (0, 1)
∼ N
X 12 + X 22 + X
X 2
(0, 1) ∼ N (0,
· · · + X n2 ∼ χ2(n).
∼ χ2(1)
χ2 (n)
n
t(n) X
∼ t(n)
n
E(X )
= 0, D2 (X ) =
X
n
n
− 2. F (m, n)
∼ F (m, n)
E(X )
=
n
−
x2 1+ n
,
x
∈ R.
m, n
f ( f (x) =
n
− n+1 2
√
Γ n+1 2 f ( f (x; n) = nπ Γ n2
m
2 Γ( m+n ) m (m −1 1 + n) 2 2 x m n Γ( 2 )Γ( 2 ) 0
2n2 (n + m 2) , D (X ) = . 2 m(n 2)2 (n 4) 2
− −
−
−
1876 1937 1890 1962
−
m n
− m+2 n
x
,
x > 0;
,
x
≤ 0.
C (λ, µ) X
∼ C (λ, µ) f ( f (x; λ, µ) =
1789
− 1857
π [(x [(x
−
λ , µ)2 + λ2 ]
x
∈ R.
...
2002 &
2008
&
2008
1968 2000 2006 2004 1985
1980 1976 2000 3 2005 2 1998 2
2002 2010 1987
2001
