x
" bt t
" vt
t
zt = bt · vt.
Z = bT (x) · vT (x).
n n
n
Å„Å‚
ôÅ‚
òÅ‚
1, t n
bt =
ôÅ‚
ół
0, t >n
vt = vt, t 0,
Å„Å‚
ôÅ‚
òÅ‚
vT (x), T (x) n
Z =
ôÅ‚
ół
0, T (x) >n
E[Z]
n
1
Ax:n|
" n
1
Ax:n| = E[Z] = zt · g(t)dt = vt · px · µx+tdt.
t
0 0
j Z
n
j
E Zj = vt · px · µx+tdt
t
0
n
= e-(´·j)t · px · µx+tdt.
t
0
j Z
n
(j · ´)
x
t ´t bt vt
bj = bt t E[Zj]
t
´t E[Z]
(j · ´t) j > 0 E[Zj]@´t = E[Z]@(j · ´t)
E Zj = E (bT (x) · vT (x))j
j
= E bj · vT (x)
T (x)
j
= E bT (x) · vT (x) .
t
vt = exp - ´sds ,
0
t
j
t
j
vt = exp - j · ´sds ,
0
(j · ´t)
2
1 1
2
Var[Z] = Ax:n| - Ax:n| ,
1
2
Ax:n| n
2´
bt = 1, t 0,
vt = vt, t 0,
Z = vT (x), T (x) 0.
"
Ax = E[Z] = vt · px · µx+tdt.
t
0
2
2
Var[Z] = Ax - Ax ,
2
Ax
2´
2 30.05.98
É = 100
(40) 100
a60| = 20
A40 a60| = 10, 6
A40
60
s(40 + t) -s (40 + t)
(A40)|100 = 100 · vt · · dt
0 s(40) s(40 + t)
60
x 100
= s(x) = 1 - = vtdt
100 60 0
100 100 100
= · a60|´ = · 20 = ,
60 60 3
60
s(40 + t) -s (40 + t)
(2A40)|100 = (100)2 · v2t · · dt
0 s(40) s(40 + t)
60
x 10.000
= s(x) = 1 - = v2tdt
100 60 0
10.000 10.000
= · a60|´ =2·´ = · 10, 6
60 60
10.000 100 2
Var(Z) = · 10, 6 - H" 655, (5),
60 3
660
n n
n n
n
Å„Å‚
ôÅ‚
òÅ‚
0, t n
bt =
ôÅ‚
ół
1, t > n
vt = vn, t 0,
Å„Å‚
ôÅ‚
òÅ‚
0, T (x) n
Z =
ôÅ‚
ół
vn, T (x) > n
1
Ax:n| = E[Z] = vn · P (T (x) > n) = vn · px
n
E[Z2] = v2n · P (T (x) > n) = v2n · px.
n
Var[Z] = v2n · px - (vn · px)2 = v2n · px · qx
n n n n
2
2 1 1
= Ax:n| - Ax:n| .
n n
n
n
n
bt = 1, t 0,
Å„Å‚
ôÅ‚
òÅ‚
vt, t n
vt =
ôÅ‚
ół
vn, t > n
Å„Å‚
ôÅ‚
òÅ‚
vT (x), T (x) n
Z =
ôÅ‚
ół
vn, T (x) > n
Ax:n|
Z1 Z2 Z3
n n
n
Å„Å‚
ôÅ‚
òÅ‚
vT (x), T (x) n
Z1 =
ôÅ‚
ół
0, T (x) > n
Å„Å‚
ôÅ‚
òÅ‚
0, T (x) n
Z2 =
ôÅ‚
ół
vn, T (x) > n
Å„Å‚
ôÅ‚
òÅ‚
vT (x), T (x) n
Z3 =
ôÅ‚
ół
vn, T (x) > n
Z3 = Z1 + Z2
1
1
Ax:n| = Ax:n| + Ax:n|.
j
E Z3 @´ = E [Z3] @(j · ´),
2
Var [Z3] =2 Ax:n| - Ax:n| .
Var[Z3] = Var[Z1 + Z2] = Var[Z1] + Var[Z2] + 2 Cov[Z1, Z2].
Z1 · Z2 = 0.
Cov[Z1, Z2] = - E[Z1] · E[Z2].
1
1
Cov[Z1, Z2] = -Ax:n| · Ax:n|.
Var [Z3] n
n
1
1
Cov[Z1, Z2] = -Ax:n| · Ax:n| = 0,

Z1 Z2
2 16.11.96
Z1 Z2 Z3
40 20 20
20
E(Z1) E(Z2)
Var(Z1) = 0, 0081 Var(Z2) = 0, 0625 Var(Z3) = 0, 0106
A40:20| = 0, 4 A40:20|
A40:20|
Var(Z3) = Var(Z1 + Z2) = Var(Z1) + Var(Z2) - E(Z1) · E(Z2)
E(Z3) = E(Z1) + E(Z2).
E(Z1) = x, E(Z2) = y.
Å„Å‚
ôÅ‚
òÅ‚
x + y = 0, 4
ôÅ‚
ół
0, 0106 = 0, 0081 + 0, 0625 - 2xy,
E(Z1) = 0, 3 E(Z2) = 0, 1 E(Z1) = 0, 1
E(Z2) = 0, 3
m
m m
m
Å„Å‚
ôÅ‚
òÅ‚
1, t > m
bt =
ôÅ‚
ół
0, t m
vt = vt, t > 0,
Å„Å‚
ôÅ‚
òÅ‚
vT (x), T (x) > m
Z =
ôÅ‚
ół
0, T (x) m
"
Ax = vt · px · µx+tdt.
m| t
m
5
µ = 0, 04
´ = 0, 10
µ ´
"
µ
Ax = e-´t · e-µt · µdt = e-5(µ+´).
5|
5 µ + ´
2
Ax = e-0,7 = 0, 1419.
5|
7
2
µ µ
Var[Z] = e-5(µ+2´) - e-5(µ+´)
µ + 2´ µ + ´
0, 04 4
= e-5(0,04+0,20) - e-1,4 = 0, 0301.
0, 04 + 0, 20 49
Z
1
P (Z = 0) = P (T (x) 5) = 1 - e-5µ = 1 - e-0,20 = 0, 1813 <
2
P (Z > 0) = P (T (x) > 5) = e-5µ = e-0,20 = 0, 8187.
Z (0, ")
P (Z meZ(x)) = P (Z = 0) + P (0 < Z meZ(x)) = 0, 5,
P (0 < Z meZ(x)) = 0, 5 - 0, 1813 = 0, 3187.
P (0 < Z meZ(x)) = P vT (x) meZ(x)
= P (T (x) · ln v ln(meZ(x)))
ln(meZ(x))
= P T (x) = 0, 3187.
ln v
ln(meZ(x))
h = Z
ln v
h
px = 0, 3187.
h
e-µh = 0, 3187,
ln(0, 3187)
h = ,
-µ
ln(meZ(x)) ln(0, 3187)
= ,
ln(e-´) -µ
´
ln(meZ(x)) = ln(0, 3187),
µ
´
µ
meZ(x) = 0, 3187 = 0, 0573.