Кошкин Г.М. Основы страховой математики
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p
1
,
p
2
, . . . , p
n
,
b
1
, b
2
, . . . , b
ν
, ν ≤ n,
ν
b
j
>> p
j
,
p
p
b, p << b.
S = b
1
+ b
2
+ ··· + b
ν
b
j
j
P(S > U),
P(S ≤ U).
P(S > U) + P(S ≤ U) = 1.
n x = 26
p,
p ·n,
U = pn. b
i
= 1
i
i = 1, . . . , n
P(b
i
= 1) = q
26
= 0, 00293, P(b
i
= 0) = p
26
= 1 − q
26
= 0, 99707.
P
(
n
X
i=1
b
i
≤ U
)
= P
n
X
i=1
b
i
− E
n
X
i=1
b
i
v
u
u
t
D
n
X
i=1
b
i
≤
U − E
n
X
i=1
b
i
v
u
u
t
D
n
X
i=1
b
i
.
Eξ Dξ
ξ.
b
i
, i = 1, . . . , n
Eb
i
= 1 · q
26
+ 0 · p
26
= q
26
= 0, 00293,
Db
i
= Eb
i
− (Eb
i
)
2
= 0, 00293 − (0, 00293)
2
≈ 0, 00292.
b
i
P
n
X
i=1
b
i
− n · 0, 00293
p
n · 0, 00292)
≤
U − n · 0, 00293
√
n · 0, 00292
.
n = 3071,
nEb
1
= 3071 · 0, 00293 ≈ 9, nDb
1
= 3071 · 0, 00292 ≈ 9,
S
S.
P
n
X
i=1
b
i
− 9
3
≤
U − 9
3
≈ Φ
U − 9
3
,
Φ(x) =
1
√
2π
Z
x
−∞
e−u
2
/2du
U − 9
3
Φ
U − 9
3
= 0, 95,
U − 9
3
= x
0,95
, x
0,95
= 1, 645
U = 3 · 1, 645 + 9 ≈ 13, 935,
b = 1,
p =
U
n
=
13, 935
3071
= 0, 00453.
b = 100000
p = 100000 · 0, 00453 ≈ 453
38
p
x = 26
S
p ≈ 114
p ≈ 130
X
X
i
i X
X, X ≥ 0.
X
F (x) = P(X ≤ x).
s(x) = P(X > x) = 1 − F (x) ,
x
x (x)
s(x)
s(x)
s(0) = 1, s(∞) = 0
s(x)
∆x = x
2
−x
1
s(x)
x
◦
∆P = s(x
◦−
) − s(x
◦
), s(x
◦−
) = lim
x→x
◦−
s(x), s(x
◦
) = lim
x→x
◦+
s(x).
x