Кошкин Г.М. Основы страховой математики
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f(t|·), s(t|·), µ
t|·
m
(x
1
, x
2
, . . . , x
m
)
k
T (x
k
) = X − x
k
.
m T (x
1
), T (x
2
), . . . , T (x
m
)
U
T (U)
U := x
1
: x
2
: . . . : x
m
(x
1
: x
2
: . . . : x
m
)
T (U) = {T(x
1
), T (x
2
), . . . , T (x
m
)}.
P{T (U) > t} = P{ (T (x
1
), T (x
2
), . . . , T (x
m
)) > t} =
= P{T (x
1
) > t, T (x
2
) > t, . . . , T (x
m
) > t},
P{T (U) > t} =
m
Y
i=1
t
p
x
i
.
t
p
x
i
= P{T (x
i
) > t}
T (U)
t
q
x
1
:x
2
:...:x
m
= 1 −
t
p
x
1
:x
2
:...:x
m
= 1 −
m
Y
i=1
(1 −
t
q
x
i
).
f
x
1
:x
2
:...:x
m
(t) = −
d
d t
P{T (U) > t} =
= −
d
d t
m
Y
i=1
t
p
x
i
= −
d
d t
m
Y
i=1
s(x
i
+ t)
s(x
i
)
.
U :=
x
1
: x
2
m = 2, T (U) = (T (x
1
), T (x
2
)),
f
x
1
:x
2
(t) =
d
dt
{−P{ (T (x
1
), T (x
2
)) > t}} =
−
s(x
1
+ t)
s(x
1
)
s(x
2
+ t)
s(x
2
)
0
t
=
==
f(x
1
+ t)
s(x
1
)
s(x
2
+ t)
s(x
2
)
+
f(x
2
+ t)
s(x
2
)
s(x
1
+ t)
s(x
1
)
=
= f
x
1
(t) s
x
2
(t) + f
x
2
(t) s
x
1
(t) < f
x
1
(t) + f
x
2
(t),
s
x
(t) =
s(x + t)
s(x)
T (x).
f
x
(t)
s
x
(t) = I
t
(−∞, ω − x) −
t I
t
(0, ω − x)
ω − x
,
f
x
1
:x
2
(t) =
I
t
(0, ω − x
1
)
ω − x
1
I
t
(−∞, ω − x
2
) −
t I
t
(0, ω − x
2
)
ω − x
2
+
+
I
t
(0, ω − x
2
)
ω − x
2
I
t
(−∞, ω − x
1
) −
t I
t
(0, ω − x
1
)
ω − x
1
=
=
ω − x
2
− t
(ω − x
1
)(ω − x
2
)
+
ω − x
1
− t
(ω − x
1
)(ω − x
2
)
I
t
(0, min(ω − x
1
, ω − x
2
)).
T (x) = X − x
µ
x
(t) =
f
x
(t)
s
x
(t)
=
f(x + t)
s(x + t)
= µ
x+t
= −
d
d t
ln s(x + t) =
= −
d
d t
[ln s(x + t) − ln s(x)] = −
d
d t
ln
s(x + t)
s(x)
= −
d
d t
ln
t
p
x
,
µ
x
(t) = µ
x+t
= −
d
d t
ln
t
p
x
µ
x
1
:x
2
:...:x
m
(t) = −
d
d t
ln
t
p
x
1
:x
2
:...:x
m
= −
d
d t
m
X
i=1
ln
t
p
x
i
=
m
X
i=1
µ
x
i
(t)
µ
x
1
:x
2
:...:x
m
(t) =
m
X
i=1
µ
x
i
(t)
µ
x
i
(t) = µ
x
i
+t
= B e
α(x
i
+t)
= B r
x
i
+t
,
r = e
α
, t ≥ 0, i = 1, . . . , m.
r
x
1
+ r
x
2
+ ··· + r
x
m
= r
ω
,
ω,
ω =
1
α
ln {r
x
1
+ r
x
2
+ ··· + r
x
m
} =
=
1
α
ln {e
αx
1
+ e
αx
2
+ ··· + e
αx
m
}.
µ
x
1
:x
2
:...:x
m
(t) = µ
ω
(t), t ≥ 0
ω,
x
1
, x
2
, . . . , x
m
ω
µ
x
i
(t) = µ
x
i
+t
= A + B e
α(x
i
+t)
= A + B r
x
i
+t
,
t ≥ 0, i = 1, . . . , m, r = e
α
.
A + B r
x
1
+t
+ ··· + A + B r
x
m
+t
= m
A + B r
ω+t
r
x
1
+ r
x
2
+ ··· + r
x
m
= m r
ω
,
ω =
1
α
[ln{r
x
1
+ r
x
2
+ ··· + r
x
m
} − ln m] =
=
1
α
[ln{e
αx
1
+ e
αx
2
+ ··· + e
αx
m
} − ln m] .
µ
x
1
:x
2
:...:x
m
(t) = m µ
ω
(t) = µ
ω:ω:...:ω
(t)
m x
1
, x
2
, . . . , x
m
m
ω,
U := x
1
: x
2
: . . . : x
m
(x
1
: x
2
: . . . : x
m
)
T (U) = (T (x
1
), T (x
2
), . . . , T (x
m
)).
t
q
x
1
:...:x
m
= P{T (U) ≤ t} = P{ (T (x
1
), T (x
2
), . . . , T (x
m
)) ≤ t} =
= P{T (x
1
) ≤ t, T (x
2
) ≤ t, . . . , T (x
m
) ≤ t},
t
q
x
1
:x
2
:...:x
m
=
m
Y
i=1
t
q
x
i
,
t
p
x
1
:x
2
:...:x
m
= 1 −
m
Y
i=1
(1 −
t
p
x
i
) .
f
x
1
:x
2
:...:x
m
(t) =
d
d t
P{T (U) ≤ t} =
d
d t
m
Y
i=1
(1 −
t
p
x
i
) .
U := x
1
: x
2
m = 2, T (U ) = (T (x
1
), T (x
2
)),
f
x
1
:x
2
(t) =
d
dt
{P(T (x
1
) ≤ t)P(T (x
2
) ≤ t)} =
d
dt
{F
x
1
(t) F
x
2
(t)} =
=
d
dt
F (x
1
+ t)
s(x
1
)
F (x
2
+ t)
s(x
2
)
=
f(x
1
+ t)
s(x
1
)
F (x
2
+ t)
s(x
2
)
+
+
f(x
2
+ t)
s(x
2
)
F (x
1
+ t)
s(x
1
)
= f
x
1
(t) F
x
2
(t) + f
x
2
(t) F
x
1
(t) < f
x
1
(t) + f
x
2
(t).
f
x
1
:x
2
(t) =
I
t
(0, ω − x
1
)
ω − x
1
t I
t
(0, ω − x
2
)
ω − x
2
+ I
t
(−∞, ω − x
2
)
+
+
I
t
(0, ω − x
2
)
ω − x
2
t I
t
(0, ω − x
1
)
ω − x
1
+ I
t
(−∞, ω − x
1
)
=
=
2t I
t
(0, min(ω − x
1
, ω − x
2
))
(ω − x
1
)(ω − x
2
)
+
+
I
t
(min(ω − x
1
, ω − x
2
), max(ω − x
1
, ω − x
2
))
min(ω − x
1
, ω − x
2
)
.
µ
x
1
:...:x
m
(t) =
f
x
1
:...:x
m
(t)
s
x
1
:...:x
m
(t)
=
d
d t
m
Y
i=1
(1 −
t
p
x
i
)
,
1 −
m
Y
i=1
(1 −
t
p
x
i
)
!
.
T (70) T (75)
5 10
(70 : 75)
P{5 < T (70 : 75) ≤ 10} = P{T (70 : 75) > 5} − P{T (70 : 75) > 10} =
=
5
p
70:75
−
10
p
70:75
=
5
p
70 5
p
75
−
10
p
70 10
p
75
=
=
l
75
l
70
l
80
l
75
−
l
80
l
70
l
85
l
75
=
l
80
l
70
1 −
l
85
l
75
.
(70 : 75)
P{5 < T (70 : 75) ≤ 10} = P{T (70 : 75) ≤ 10} − P{T (70 : 75) ≤ 5} =
=
10
q
70:75
−
5
q
70:75
=
10
q
70 10
q
75
−
5
q
70 5
q
75
=
= (1 −
10
p
70
)(1 −
10
p
75
) − (1 −
5
p
70
)(1 −
5
p
75
) =
=
1 −
l
80
l
70
1 −
l
85
l
75
−
1 −
l
75
l
70
1 −
l
80
l
75
.
P
m
{5 < T (70 : 75) ≤ 10} =
18, 787
43, 405
1 −
9, 063
30, 857
= 0, 3057,
P
m
{5 < T (70 : 75) ≤ 10} =
1 −
18, 787
43, 405
1 −
9, 063
30, 857
−
−
1 −
30, 857
43, 405
1 −
18, 7870
30, 857
= 0, 2875
P
m1
= P{5 < T (70) ≤ 10} =
l
75
l
70
−
l
80
l
70
=
30, 857
43, 405
−
18, 787
43, 405
= 0, 2781,
P
m2
= P{5 < T (75) ≤ 10} =
l
80
l
75
−
l
85
l
75
=
18, 787
30, 857
−
9, 063
30, 857
= 0, 3151,
P
w
{5 < T (70 : 75) ≤ 10} =
41, 674
70, 043
1 −
24, 265
57, 679
= 0, 3447,
P
w
{5 < T (70 : 75) ≤ 10} =
1 −
41, 674
70, 043
1 −
24, 265
57, 679
−
−
1 −
57, 679
70, 043
1 −
41, 674
57, 679
= 0, 1856,
P
w1
= P{5 < T (70) ≤ 10} =
57, 679
70, 043
−
41, 674
70, 043
= 0, 2285,
P
w2
= P{5 < T (75) ≤ 10} =
41, 674
57, 679
−
24, 265
57, 679
= 0, 30182.
U := x
1
: x
2
: . . . : x
m
U := x
1
: x
2
: . . . : x
m
,
◦
e
U
= ET (U) =
Z
∞
0
t f
U
(t) dt =
Z
∞
0
t
p
U
dt,
DT (U) =
Z
∞
0
t
2
f
U
(t) dt −
◦
e
U
2
= 2
Z
∞
0
t
t
p
U
dt −
◦
e
U
2
.
◦
e
x
1
:x
2
◦
e
x
1
:x
2
=
Z
∞
0
t f
x
1
:x
2
(t) dt =
Z
min(ω−x
1
,ω−x
2
)
0
t
2t
(ω − x
1
)(ω − x
2
)
dt+
+
Z
max(ω−x
1
,ω−x
2
)
min(ω−x
1
,ω−x
2
)
t
dt
(ω − x
1
)(ω − x
2
)
=
=
2 min
3
(ω − x
1
, ω − x
2
)
3(ω − x
1
)(ω − x
2
)
+
+
max
2
(ω − x
1
, ω − x
2
) − min
2
(ω − x
1
, ω − x
2
)
2 max(ω − x
1
)(ω − x
2
)
=
=
2 min
2
(ω − x
1
, ω − x
2
)
3 max(ω − x
1
)(ω − x
2
)
+
+
max
2
(ω − x
1
, ω − x
2
) − min
2
(ω − x
1
, ω − x
2
)
2 max(ω − x
1
)(ω − x
2
)
=
=
min
2
(ω − x
1
, ω − x
2
) + 3 max
2
(ω − x
1
, ω − x
2
)
6 max(ω − x
1
)(ω − x
2
)
.
◦
e
x:x
◦
e
x:x
= 2
Z
ω−x
0
t
ω − x − t
(ω − x)
2
)
dt =
ω − x
3
.
k
k k
U :=
k
x
1
: x
2
: . . . : x
m
k m
(x
1
), (x
2
), . . . , (x
m
),
(m − k + 1)
m
x
1
: x
2
: . . . : x
m
= (x
1
: x
2
: . . . : x
m
),
1
x
1
: x
2
: . . . : x
m
= (x
1
: x
2
: . . . : x
m
),
k = m
k = 1
k [k]
U :=
[k]
x
1
: x
2
: . . . : x
m
k m (x
1
), (x
2
), . . . , (x
m
),
(m −k)
(m − k + 1)
k
(x
1
: x
2
: x
3
: x
4
) ;
x
1
: x
2
: (x
3
: x
4
)
; (x
1
: x
2
: x
3
: x
4
) .
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
)}}.
x
3
x
4
x
1
x
2
x
1
: x
2
: (x
3
: x
4
)
T (U) = { {T (x
1
), T (x
2
)}, {T (x
3
), T (x
4
)}}.