Кошкин Г.М. Основы страховой математики
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x
1
x
2
x
3
x
4
(x
1
: x
2
: x
3
: x
4
)
T (U) = {T(x
1
), T (x
2
), {T (x
3
), T (x
4
)}}.
U := x
1
: x
2
U := x
1
: x
2
f(t) = e
−t
, t ≥ 0,
f
x
1
:x
2
(t) = 2 e
−2t
, f
x
1
:x
2
(t) = 2 e
−t
(1 − e
−t
), t ≥ 0.
T (65), T (70) T (75)
5 10
◦
e
x
1
:x
2
.
◦
e
x:x
=
2(ω − x)
3
, DT (x : x) = DT (x : x) =
(ω − x)
2
18
.
f(t) = λe
−λ t
, λ > 0, t ≥ 0,
f
x
1
:x
2
:x
3
(t), f
x
1
:x
2
:x
3
(t), f
x
1
:x
2
:x
3
(t),
◦
e
x
1
:x
2
:x
3
,
◦
e
x
1
:x
2
:x
3
,
◦
e
x
1
:x
2
:x
3
.
x
1
: x
2
: (x
3
: x
4
) : x
5
;
x
1
: (x
2
: x
3
) : (x
4
: x
5
) : x
6
.
♠
t
q
x
,
t
p
x
, q
x
, p
x
,
t|u
q
x
,
t|
q
x
.
A A
1
, A
2
, . . . , A
N
A
P(A), 1 −
P(A). P(A)
P
N
(A) =
1
N
N
X
i=1
I(A
i
) =
k
N
,
I(A) :
I(A) =
1, ω ∈ A,
0, ω /∈ A,
ω
k
N
A N
P
N
(A)
P
N
(A) P(A),
E P
N
(A) = P(A);
D P
N
(A) =
1
N
P(A)(1 − P(A)) <
1
N
→ 0 N → ∞;
N P
N
(A)
P{N P
N
(A) = k} = C
k
N
[P(A)]
k
[1 − P(A)]
N−k
;
P
N
(A) P(A),
lim
N→∞
P{|P
N
(A) − P(A)| < ε} = 1 ε > 0;
P
N
(A)
N
X
i=1
c
i
I(A
i
), c
i
> 0;
P
N
(A) 1 P(A),
P{ lim
N→∞
|P
N
(A) − P(A)| = 0} = 1.
5
b
P(A) = P
N
(A)
"
1 +
b
D P
N
(A)
P
2
N
(A)
#
−1
= P
N
(A)
1 +
1 − P
N
(A)
P
N
(A) N
−1
D P
N
(A)
u
2
b
P(A)
= D
b
P(A) + b
2
(
b
P(A)) < D P
N
(A),
b(
b
P(A)) = E
b
P(A) − P(A)
b
P(A).
b
P(A) P
N
(A)
F (t, λ) = 1 − e
−λt
, t ≥ 0.
λ
F (t, λ, α) = 1 − e
−(λt)
α
,
t ≥ 0, λ, α > 0,
=⇒
N
s
{µ, σ} s
µ = (µ
1
, . . . , µ
s
)
σ =
"
σ
11
··· σ
1s
··· ··· ···
σ
s1
··· σ
ss
#
, 0 ≤ σ
ij
= σ
ij
(x) < ∞, i, j = 1, s.
F (x) = P(X ≤ x), s(x) = P(X < x), N
{X
i
, i =
1, N}, N
F
N
(x) =
1
N
N
X
i=1
I(X
i
≤ x), s
N
(x) =
1
N
N
X
i=1
I(X
i
> t).
F
N
(x)
F (x)
F
N
(x)
P
N
(A).
EF
N
(x) = F (x);
DF
N
(x) =
1
N
F (x)(1 − F (x));
F
N
(x) = F (x) +
ξ
N
(x)
√
N
,
ξ
N
(x)
{0, F (x) (1 − F (x))}
ξ
N
(x) =
1
√
N
N
X
i=1
[I(X
i
≤ x) − F (x)] =⇒ N
1
{0, F (x) (1 − F (x))}.
F
N
(x)
N
X
i=1
c
i
I(X
i
≤ x), c
i
> 0.
s
N
(x) =
1
N
N
X
i=1
I(X
i
> x) =
1
N
N
X
i=1
[1 − I(X
i
≤ x)] = 1 − F
N
(x),
s
N
(x)
F
N
(x).
Es
N
(x) = s(x);
Ds
N
(x) =
1
N
F (x)(1 − F (x)) =
1
N
s(x)(1 − s(x)).
F
N
(x) s
N
(x)
X
1
, . . . , X
n
F
N
(x) = 0 Ω
0
= [0, min (X
1
, . . . , X
N
)] , s
N
(x) = 0
Ω
∞
= [max (X
1
, . . . , X
N
) , ∞] .
cµ
x
=
f
N
(x)
s
N
(x)
,
Ω
∞
f
N
(x)
S(u)
Suv S(u) ∈ Suv S(u)
S(u) : R
1
→
R
1
S(−∞) = 1, S(∞) = 0.
s
N
(x)
fs
N
(x) =
1
N
N
X
i=1
S
x − X
i
a
N
,
S(u) ∈ Suv a
N
↓ 0. S(u)
fs
N
(x)
a
N
= 0
s
N
(x) fs
N
(x)|
a
N
=0
= s
N
(x).
H(z) : R
s
→ R
1
N
ν,s
(x) H(z)
ν x; H(z) ∈ N
ν,s
(R),
H(z) x ∈ R
s
.
s(x)
fs
N
s(z) ∈ N
0,1
(x), s(z) x, S(u) ∈ Suv,
a
N
↓ 0,
lim
N→∞
E fs
N
(x) = s(x).
s(·) x
E fs
N
(x) = E
"
1
N
N
X
i=1
S
x − X
i
a
N
#
=
Z
R
1+
S
x − y
a
N
dF (y) =
=
x
Z
0
S
x − y
a
N
dF (y) +
∞
Z
x
S
x − y
a
N
dF (y),
R
1+
= [0, ∞).
lim
N→∞
S
x − y
a
N
=
0, y < x,
1, y > x,
lim
N→∞
Z
R
1+
S
x − y
a
N
dF (y) =
∞
Z
x
dF (y) = 1 − F (x) = s(x). ♠
fs
N
(x) u
2
(fs
N
(x)) =
E (fs
N
(x) − s(x))
2
s(x) (1 − s(x))
N
s
N
(x)
b (fs
N
(x)) = E fs
N
(x) − s(x).
fs
N
(x)|
a
N
=0
= s
N
(x).
S(u)
fs
N
(x) fs
N
(x)
S
LAP
(u) = 1 − F
LAP
(u) F
LAP
(u)
F
LAP
(u) =
0.5e
u
, −∞ < u < 0,
1 − 0.5e
−u
, 0 ≤ u < ∞.
S
LAP
(u) =
1 − 0.5e
u
, −∞ < u < 0,
0.5e
−u
, 0 ≤ u < ∞.
fs
N
LAP
(x) =
1
N
N
X
i=1
S
LAP
x − X
i
a
N
.
fs
N
LAP
s(z) ∈
N
0,1
(x), sup
x∈R
1+
f(x) ≤ C < ∞, a
N
↓ 0. N → ∞
|b (fs
N
LAP
(x))| = O (a
N
) .
E fs
N
LAP
(x) =
Z
R
1+
S
LAP
x − y
a
N
dF (y) =
= s(x) +
x
Z
0
S
LAP
x − y
a
N
dF (y) +
∞
Z
x
S
LAP
x − y
a
N
− 1
dF (y) =
= s(x) +
x
Z
0
0.5e
−
x−y
a
N
f(y)dy +
∞
Z
x
h
1 − 0.5e
x−y
a
N
− 1
i
f(y)dy.
u =
x − y
a
N
|b (fs
N
LAP
(x))| ≤ C
a
N
2
x/a
N
Z
0
e
−u
du +
0
Z
−∞
e
u
du
=
= C
a
N
2
−e
−x/a
N
+ 1 + 1 − e
−∞
= O (a
N
) . ♠
fs
N
(x)
N → ∞
∞
Z
x
S
x − y
a
N
− 1
dy,
x
Z
0
S
x − y
a
N
dy.
S(u) ∈ Suv,
S(u) =
1 − F(u), F(u)
S(u) =
1
2
−
arctan(u)
π
,
S(u)
F in
S
S(u) ∈ F in
S
S(u) =
(
1, −∞ < u < C
1
,
Z(u), C
1
≤ u ≤ C
2
,
0, C
2
< u < ∞,
Z(u)
Z(C
1
) = 1 Z(C
2
) = 0 C
1
< C
2