K(x)
Å„Å‚
ôÅ‚
1
òÅ‚
P (K(x) m(x)) ,
2
ôÅ‚
1
ół
P (K(x) m(x)) .
2
3.13
1600
10 10 540
x lx
0 40
10 39
70 26
Y1 Y2 70
c
P (Y1 + Y2 >c) = 0, 05.
Ti(x)
39
t t
l10 = s(10) · l0 = s(10) · 40 Ò! s(10) = .
40
540 · 40
2 2 2
l10 = s(10) · l0 Ò! l0 = H" 554.
39
2 1
l0 = 554 l0 = 1600 l0 = 2154
Y1 + Y2 70
Y = Y1 + Y2 (l0, s(70))
t
l70 13
13
s(70) = = E[Y ] = l0 · s(70) = · 2154 Var[Y ] =
t
l0 20 20
13 7
l0 · s(70) · (1 - s(70)) = · · 2154
20 20
ëÅ‚ öÅ‚
13
c
(Y1 + Y2) - E(Y1 + Y2) - · 2154
20
íÅ‚ Å‚Å‚
P (Y1 + Y2 > c) = P > = 0, 05.
13 7
Var(Y1 + Y2) · · 2154
20 20
ëÅ‚ öÅ‚
13
c - · 2154
20
íÅ‚ Å‚Å‚
1 - Åš = 0, 95
13 7
· · 2154
20 20
13
c - · 2154
20
H" 1, 645 Ò! c H" 1436, 4136.
13 7
· · 2154
20 20
T (x)
s(x)
x " {0, 1, . . .}, 0 t 1
s(x + t) = (1 - t)s(x) + ts(x + 1)
x
s(x + t) = s(x)e-µ t µx = - ln px
1 1-t t
= +
s(x+t) s(x) s(x+1)
t·qx
x
qx t · qx 1 - e-µ ·t
t
1-(1-t)·qx
px
x
px 1 - t · qx e-µ ·t
t
1-(1-t)·qx
y·qx y·qx
x
qx+t 1-t·qx 1 - e-µ ·y
y
1-(1-y-t)·qx
qx qx
µx+t 1-t·qx µx
1-(1-t)·qx
px·qx
x
px · µx+t qx e-µ ·t · µx
t
[1-(1-t)·qx]2
0 t = 0 t = 1
s(x+t) (1-t)s(x)+ts(x+1)
" t
qx = 1 - = | | = 1 -
s(x) s(x)
s(x+1)
= 1 - (1 - t) - ts(x+1) = t - ts(x+1) = t 1 - = t · qx
s(x) s(x) s(x)
s(x+t+y) (1-t-y)s(x)+(t+y)s(x+1)
" y
qx+t = 1 - = | | = 1 -
s(x+t) (1-t)s(x)+ts(x+1)
s(x+1)
y 1- ]
[
ys(x)-ys(x+1) s(x) y·qx
= = =
s(x+1)
s(x)-t[s(x)-s(x+1)] 1-t·qx
1-t 1- ]
[
s(x)
s(x+1)
1-
s(x)-s(x+1) s(x)
" µx+t = -s (x+t) = | | = =
s(x+1)
s(x+t) (1-t)s(x)+ts(x+1)
1-t 1- )
(
s(x)
qx
=
1-t·qx
3.24
3.2 55
T (x) x = 0
T (0)
1
P (T (0) > m(0)) =
2
lm(0) 1
s(m(0)) = = .
l0 2
3.2 l77 = 51599 l78 = 48878
(77; 78)
s(77 + t) = (1 - t) · s(77) + t · s(78) = 0, 5
(1 - t) · l77 + t · l78 = 50000.
t = 0, 5876516.
m(0) = 77, 5876516.
T (x) = K(x) + S(x),
S(x)
x k
P (k < T (x) k + s) = P (K(x) = k )" S(x) s)
= qx
k|s
= px · qx+k.
k s
P (K(x) = k )" S(x) s) = px · s · qx+k
k
= s · qx
k|
= P (K(x) = k) · P (S(x) s).
K(x) S(x)
P (S(x) s) = s 0 < s < 1,
S(x) (0, 1)
P (X > x + t|X > x) = P (T (x) > t)
[x]
r
"j > 0q[x-j]+r+j H" q[x]+r = qx+r.
r
3.7 p[30] 3q[31]+1
2
l[30]+2
l32 33795
p[30] = = | r = 2| = = = 0, 99899
2
l[30] l[30] 33829
l[31]+1+3
l35 33719
q[31]+1 = 1 - = | r = 2| = 1 - = 1 - = 0, 00213.
3
l[31]+1 l[31]+1 33791