1.1 (3) xf2ASBxdxa4x70xd8x83x2x1bx38xdcx1bxbfxb5ASB = AS(B?A) = ASB?ABxa7x32x64
x56xc7x1bx8cx5cx35x8cx1ax22
(4) xcfx8fB ‰ Axa7xa4xb1A = BSA?Bxa7x64x64x8cx1ax22
1.3 x8ATB 6= `x9exa7 (A)x1x6b16x87x3x83x22x8ATB = `x9exa7 (A)x6b8x87x3x83x22xd8x4ax15xd1
1.4 x4bx38xa5A3 = f(a;b),a;b 2 Rg
(1) xcfx8f8n;A ‰ Anxa4xb1x6bA ‰
1\
n=1
Anx22
x65xa1x79xb2
1\
n=1
An ‰ Ax22x87x79x7bxb5x62x179x 2
1\
n=1
An;x =2 Axa7x40x6fx? a > 0xa7x12n = d 1x?ae + 1xa7
x4bx > a + 1nxa7x64x64x8cx1ax =2 Anxa7x67xf1x22xa4xb1
1\
n=1
An ‰ Ax22x64x64x8cx1aA =
1\
n=1
Anx22x61x71x8c
x1aB =
1[
n=1
Bnx22
(2) xcfx8f8a 2 R;(?1;a] =
1[
n=1
(?n;a]xa7xa4xb1A1 ‰ (A2)x22xcfx8f8a;b 2 R;(a;b] = (?1;b]?
(?1;a]xa7xa4xb1A2 ‰ (A1)x22x6exfex8cx1a (A1) = (A2)
(3)
8a 2 R;(?1;a] =
1[
n=1
(?n;a]
=
1\
m=1
(?1;a+ 1m)
=
1\
m=1
1[
n=1
(?n;a+ 1m)
=
1\
m=1
1[
n=1
[?n;a+ 1m)
=
1[
n=1
[?n;a]
8a;b 2 R;(a;b] = (?1;b]?(?1;a]
(a;b) =
1[
n=1
(?1;b? 1n]?(?1;a]
[a;b) =
1[
n=1
(?1;b? 1n]?
1[
m=1
(?1;a? 1m]
[a;b] = (?1;b]?
1[
m=1
(?1;a? 1m]
x64x64x8cx1a (A1) = (Ai);2? i? 5
1.6 (3) xcfx8fA2k+1 ‰ A2kxa7xa4xb1
1[
k=n
Ak =
‰ S1
k=n=2 A2k = (
1
4;
1
2 +
1
n] nx8fxf3xeaS1
k=(n+1)=2 A2k = (
1
4;
1
2 +
1
n+1] nx8fxdbxea
1
)
1\
n=1
1[
k=n
Ak =
1\
n=1
(14; 12 + 1n] = (14; 12]
x64x64x8cx1aP
1\
n=1
1[
k=n
Ak
!
= F(12)?F(14)x22
1\
k=n
Ak = ` )
1[
n=1
1\
k=n
Ak = `
x64x64x8cx1aP
1[
n=1
1\
k=n
Ak
!
= 0x22
1.10 (1) x64x34x81x1bxbdxc2
! 2f!,limn!1Xn = Xg
,8m? 1 9n? 1 8k? n jXk(!)?Xj < 1=m
,8m? 1 9n? 1 ! 2
\
k?n
f!,jXk?Xj < 1=mg
,8m? 1 ! 2
[
n?1
\
k?n
f!,jXk?Xj < 1=mg
,! 2
\
m?1
[
n?1
\
k?n
f!,jXk?Xj < 1=mg
1.14
EN =
1X
n=1
nP(N = n) =
1X
n=1
nX
m=1
P(N = n)
=
1X
n=1
1X
m=n
P(N = m) =
1X
n=1
P(N > n)
EX =
Z 1
0
xdF(x) =
Z 1
0
(
Z x
0
dy)dF(x)
=
Z 1
0
Z 1
y
dF(x)dy =
Z 1
0
(1?F(x))dx
E(Xn) =
Z 1
0
xndF(x) =
Z 1
0
(
Z x
0
nyn?1dy)dF(x)
=
Z 1
0
Z 1
y
dF(x)nyn?1dy =
Z 1
0
nxn?1(1?F(x))dx
1.18
P(? = k) =
1X
n=k
P(N = n)P(? = kjN = n)
=
1X
n=k
P(N = n)P(
nX
i=1
Xi = kjN = n)
=
1X
n=k
P(N = n)P(
nX
i=1
Xi = k)
2
=
1X
n=k
ne
n! C
k
np
k(1?p)n?k
=
1X
n=k
ne
k!(n?k)!p
k(1?p)n?k
= e
kpk
k!
1X
n=k
n?k
(n?k)!(1?p)
n?k
=
1X
n=k
ne
k!(n?k)!p
k(1?p)n?k
= e
kpk
k! e
(1?p)
= (?p)
k
k! e
p? Po(?p)
xa4xb1E(?) =?p; D(?) =?px22
1.19 (1)
P(N1 +N2 = n) =
nX
k=0
P(N1 +N2 = n;N1 = k)
=
nX
k=0
P(N2 = n?k)P(N1 = k)
=
nX
k=0
k1e1
k!
n?k2 e2
(n?k)!
= (?1 +?2)
n
n! e
(?1+?2)
(2)
P(N1 = kjN1 +N2 = n) = P(N1 = k)P(N2 = n?k)P(N
1 +N2 = n
= Ckn
1
1 +?2
k?
2
1 +?2
n?k
(3)
P(N1 +N2jN3) = P(N1jN3)+P(N2jN3) = P(N1)+P(N2) = P(N1 +N2)
xa4xb1N1 +N2x86N3xd5xe1x22
(4)
E(N1jN1 +N2 = n) =
nX
k=0
kP(N1 = kjN1 +N2 = n)
=
nX
k=0
kCkn
1
1 +?2
k?
2
1 +?2
n?k
= n?1?
1 +?2
nX
k=1
Ck?1n?1
1
1 +?2
k?1?
2
1 +?2
n?k
= n?1?
1 +?2
3
xa4xb1E(N1jN1 +N2) =?1?
1 +?2
(N1 +N2)x22
E(N1 +N2jN1) = E(N1jN1)+E(N2jN1) = N1 +E(N2) = N1 +?2
1.20 xf9x70x90x89xd1x31x6ex2x4bx1bx79xb2xa7x63xfcx87x61x71x8cx1ax22
(3)
E(E(IAjIB)jIB;IC) = E(P(AjB)IB +P(AjB)IBjIB;IC)
= E(P(AjB)IBjIB;IC)+E(P(AjB)IBjIB;IC)
= P(AjB)IB +P(AjB)IB
= E(IAjIB)
1.24 X;Yx1bxe9xdcx56xc7x97xddxbcxeax8fxb5
f(X;Y)(x;y) = e?(x+y)I(x>0;y>0)
x64 X = UV=(1+V);Y = U=(1+V) x1aJacobixddxax1bx31xfxaax8fxb5
J =?u=(1+v)2
x64x64x8cx1axb5
f(U;V)(u;v) = e?u u(1+v)2I(u > 0;v > 0)
xa9x4fxc8xa9x8cx1ax3exexa9xd9x8fxb5
fU(u) = ue?uI(u > 0)
fV (v) = 1(1+v)2I(v > 0)
1.25
P(Z? z) = P(X +Y? z)
=
nX
k=0
P(X +Y? z;X = k)
=
nX
k=0
P(Y? z?k)P(X = k)
x64x64x8cx1aZx1bxa9xd9xbcxeaxa7x32xe9xa9xd9xbcxeaxa6x13x8cx1ax56xc7x97xddxbcxeax22
1.28
E(E(XjY;Z)jY)
=
X
i
E(
X
j;k
E(XjY = yj;Z = zk)I(Y=yj)I(Z=zk)jY = yi)I(Y=yi)
=
X
j;k
E(XjY = yj;Z = zk)E(I(Z=zk)jY = yj)I(Y=yj)
xcfx8fx8i 6= jx9eE(I(Y=yj)I(Z=zk)jY = yj) = 0
=
X
j;k
X
i
xiP(X = xijY = yj;Z = zk)P(Z = zkjY = yj)I(Y=yj)
4
=
X
j;k
X
i
xiP(X = xi;Z = zkjY = yj)I(Y=yj)
=
X
j
X
i
xiP(X = xijY = yj)I(Y=yj)
=
X
j
E(XjY = yj)I(Y=yj)
= E(XjY)
x2cx98x87x61x71x8cx1ax22
2.2 x79xb2xb5
P(N(s) = kjN(t) = n)
= P(N(s) = k;N(t) = n)P(N(t) = n)
= P(N(s) = k;N(t)?N(s) = n?k)P(N(t) = n)
= P(N(s) = k)P(N(t?s) = n?k)P(N(t) = n)
=
(?s)k
k! e
s(?(t?s))n?k
(n?k)! e
(t?s)
(?t)k
k! e
t
= Ckn(st)k(1? st)n?k (0? k? n)
2.3 x29xb5(1)
E(N(t)N(s+t))
= E(N(t)2 +N(t)(N(s+t)?N(t)))
= E(N(t)2)+E(N(t))E(N(s))
= (?t)2 +?t+?2ts
(2)
E(N(s+t)jN(s))
= E(N(s))+E(N(s+t)?N(s)jN(s))
= N(s)+E(N(t)jN(s))
= N(s)+?t
P(E(N(s+t)jN(s)) = n+?t) = P(N(s) = n) = (?s)
n
n! e
s
2.5 x29xb5x17Ui? U[0;t]; i = 1;2;￠￠￠;nxa7x4bxd9x5ex53xdax4fxfex86S1;S2;￠￠￠;Snx33
N(t) = n x1bx5ex87x65x1bxa9xd9x83xd3x22
E(SkjN(t) = n) = E(U(k)) = ktn+1; (k? n)
5
2.6 x29xb5x8x? 0x9e
P(W(t)? x) = P(N(t+x)?N(t) > 0)
= 1?P(N(t+x)?N(t) = 0)
= 1?ex
x8x < 0x9exa7P(W(t)? x) = 0x22
xa4xb1P(W(t)? x) = 1?e?xIx?0x22
x80 < x < tx9e
P(V(t)? x) = P(N(t)?N(t?x) > 0)
= 1?P(N(t)?N(t?x) = 0)
= 1?ex
x8x? tx9exa7P(V(t)? x) = 1x22
x8x? 0x9exa7P(V(t)? x) = 0x22
xa4xb1
P(V(t)? x)
8<
:
1?e?x ; 0 < x < t
1 ; x? t
0 ; x? 0
x8x > 0;0 < y < tx9e
P(W(t)? x;V(t)? y)
= P(N(t+x)?N(t) > 0;N(t)?N(t?y) > 0)
= P(N(t+x)?N(t) > 0)P(N(t)?N(t?y) > 0)
= (1?ex)(1?ey)
x8y? tx9e
P(W(t)? x;V(t)? y)
= P(N(t+x)?N(t) > 0)
= P(N(t+x)?N(t) > 0)
= 1?ex
2.10 x29xb5x64x4bxbfx7f? = 1=2xa7x40x6f
P(Sn > x) =
Z 1
x
(?t)n?1
(n?1)! e
tdt
= 12n(n?1)!
Z 1
x
tn?1e?t=2dt
2.15 x29xb5x64xc4x1dx8dx23xbdx6e
limx!1 E(N(x))x = 1E(X +Y) = 23
2.16 x29xb5x2dYi = Xi?– xa7x4bYi? Ex(‰)xa7xa4xb1x64Yix29xa4x1bxb4xd1x74x36x22
P(N(t)? n)
6
= P(Sn? t)
= P(
nX
i=1
Xi? t)
= P(
nX
i=1
Yi? t?n–)
= 1?
n?1X
k=0
[?(t?n–)]k
k! e
(t?n–)
2.17 x29xb5
F(s) =
Z 1
0
e?st?2tetdt
=?
2
(s+?)2
)?m(s) =
F(s)
1F(s)
=?
2
s(s+2?)
) m0(t) =?22e?2?t
) m(t) = (?2t+ 14e?2?t? 14)I(t?0)
2.18 x29xb5x80 < x < tx9e
P(flt? x) = P(W(t)+V(t)? x)
=
Z x
0
P(W(t)? x?yjV(t) = y)dP(V(t)? y)
=
Z x
0
P(W(t)? x?y)dP(V(t)? y)
=
Z x
0
(1?e(x?y))eydy
= 1?exxex
x8x? tx9e
P(flt? x) = P(W(t)+V(t)? x)
=
Z x
0
P(W(t)? x?yjV(t) = y)dP(V(t)? y)
=
Z x
0
P(W(t)? x?y)dP(V(t)? y)
=
Z t?
0
(1?e(x?y))eydy +
Z 1
t?
P(W(t)? x?yjV(t) = y)dP(V(t)? y)
= 1?ettet +(1?e(x?t))et
= 1?extex
2.25 x29xb5
7
(1) S1;S2?S1;￠￠￠;Sn?Sn?1xd3xa9xd9x2xd8xb4x5ex87xd5xe1x22
(2) x8n = 0x9e
E(S1jN(t) = 0) = E(W(t)+tjN(t) = 0)
= t+E(W(t))
= t+ 1?
x8n? 1x9e
E(S1jN(t) = n) = E(U(1))
= tn+1
(3) x8n? kx9e
E(SkjN(t) = n) = E(xk +xk?1 +￠￠￠+xn+2 +W(t)+tjN(t) = n)
= t+ k?n?
x8n? kx9e
E(SkjN(t) = n) = E(U(k))
= ktn+1
(4) x86U(i);U(k)x1bxe9xdcxa9xd9x83xd3xa7x8cx5ex87x3x7bxbdxf6xc8xa9x1ax14x22
2.26 x29xb5
(1)
f(x;y) = lim
h!0
P(x < S2? x+h;y < S5? y +h
h2
= lim
h!0
P(N(x) = 1;N(x+h)?N(x) = 1;N(y)?N(x+h) = 2;N(y +h)?N(y = 1))
h2
= lim
h!0
xex?heh?2(y?x?h)22 e(y?x?h)?heh
h2
=?
5x(y?x)2ey
2
(2)
E(S1) = E(S1jN(t)? 1)P(N(t)? 1)+E(S1jN(t) = 0)P(N(t) = 0)
) E(S1jN(t)? 1)) = E(S1)?E(S1jN(t) = 0)P(N(t) = 0)P(N(t)? 1)
E(S1jN(t)? 1)) = E(S1)?E(S1jN(t) = 0)P(N(t) = 0)P(N(t)? 1)
=
1
(t+
1
)e
t
1?et
8
(3)
f(x;y) = lim
h!0
P(x < S1? x+h;y < S2? y +h;N(t) = 1
h2
= lim
h!0
P(N(x) = 0;N(x+h)?N(x) = 1;
N(t)?N(x+h) = 0;N(y)?N(t) = 0;N(y +h)?N(y = 1))=h2
= lim
h!0
ex?hehe(t?x?h)e(y?h?t)?heh
h2
=?t e(y?t)
2.29 x29xb5
(1) x8n = 0x9exa7
x65x? t; P(S1? x;N(t) = 0) = 0x22
x65x > t
P(S1? x;N(t) = 0)
= P(S1? xjN(t) = 0)P(N(t) = 0)
= P(W(t)? x?tjN(t) = 0)P(N(t) = 0)
= P(W(t)? x?t)P(N(t) = 0)
= (1?e(x?t))et
= et?ex
x8n? 1x9e
x65x < 0; P(S1? x;N(t) = n) = 0x22
x65x? 0
P(S1? x;N(t) = n)
= P(S1? xjN(t) = n)P(N(t) = n)
= P(U(1)? x)P(N(t) = n)I(x?t) +P(N(t) = n)I(x>t)
= [1?
t?x
t
n
](?t)
net
n! I(x?t) +
(?t)net
n! I(x>t)
(2) x8n = 0x9exa7
x65x? t; P(S2? x;N(t) = 0) = 0x22
x65x > t
P(S2? x;N(t) = 0)
= P(N(t) = 0;N(x)?N(t)? 2)
= P(N(t) = 0)P(N(x)?N(t)? 2)
= et?ex(x?t)ex
x8n = 1x9e
x65x? t; P(S2? x;N(t) = 1) = 0x22
x65x > t
P(S2? x;N(t) = 1)
= P(S2? xjN(t) = 1)P(N(t) = 1)
= P(W(t)? x?tjN(t) = 1)P(N(t) = 1)
= P(W(t)? x?t)P(N(t) = 1)
= (1?e(x?t))?tet
9
x8n? 2x9e
x65x? 0; P(S2? x;N(t) = n) = 0x22
x65x > 0
P(S2? x;N(t) = n)
= P(S2? xjN(t) = n)P(N(t) = n)
= P(U(2)? x)P(N(t) = n)I(x?t) +P(N(t) = n)I(x>t)
=
"
1?n
t?x
t
n?1 x
t?
t?x
t
n# (?t)net
n! I(x?t) +
(?t)net
n! I(x>t)
= ￡tn?n(t?x)n?1x?(t?x)n? (?t)
net
n! I(x?t) +
(?t)net
n! I(x>t)
3.1 (1)
E(X2) = 1￡ 19 +2￡ 29 +3￡ 23 = 239
E(X2jX1 = i) =
3X
j=1
j P(X2 = jjX1 = i) =
3X
j=1
j Pij
)
8<
:
E(X2jX1 = 1) = 1
E(X2jX1 = 2) = 73
E(X2jX1 = 3) = 83
) E(X2jX1) = I(X1=1) + 73I(X1=2) + 83I(X1=3)
xd3x6eE(X3jX2) = I(X2=1) + 73I(X2=2) + 83I(X2=3)
(2)
P(T = 1jX0 = 3) = P(X1 = 1jX0 = 3) = P31 = 0
P(T = 2jX0 = 3) = P(X2 = 1;X1 6= 1jX0 = 3) =
3X
i=2
P3iPi1 = 19
P(T = 3jX0 = 3) = P(X3 = 1;X1 6= 1;X2 6= 1jX0 = 3)
=
3X
i=2
3X
j=2
P(X3 = 1;X2 = i;X1 = jjX0 = 3)
=
3X
i=2
3X
j=2
P3jPjiPi1 = 227
P(T? 4jX0 = 3) = 1?
3X
k=1
P(T = kjX0 = 3) = 2227
E(T ^4jX0 = 3) =
3X
i=1
i P(T = ijX0 = 3)+4P(T? 4jX0 = 3)
= 10027
(3)
P(T11 = 1jX0 = 1) = P(X1 = 1jX0 = 1) = 0
P(T11 = 2kjX0 = 1) = P12(P23P32)k?1P21 +P13(P32P23)k?1P31
10
= 13
4
9
k?1
P(T11 = 2k +1jX0 = 1) = P12(P23P32)k?1P23P31 +P13(P32P23)k?1P32P21
= 29
4
9
k?1
) ET11 =
1X
k=1
2k13
4
9
k?1
+
1X
k=1
(2k +1)29
4
9
k?1
= 4
3.2 x79xb2xb5x5exeaxc6x38x42x7bx22
n = 1x9exa7xb7x4bx77x2cxa4xe1x22
x62x17n = kx9exb7x4bxa4xe1x22
x40x6fx8n = k +1x9exa7
Pk+1 = Pk P
= 1a+b
b a
b a
1?a a
b 1?b
+
(1?a?b)k
a+b
a?a
b b
1?a a
b 1?b
= 1a+b
b a
b a
+ (1?a?b)
k
a+b (1?a?b)
a?a
b b
= 1a+b
b a
b a
+ (1?a?b)
k+1
a+b
a?a
b b
x64xeaxc6x38x42x7bxb7x4bx1ax79x22
3.3 x79xb2xb5
`ij(z) =
1X
n=0
P(n)ij zn =
1X
n=0
P nijzn
) `(z) =
1X
n=0
zPn = (I?zP)?1
xd9xa5Pnij xb4x90xaPnx1bx31ix31x31jxfx1bx3x83x22
3.6
f(n)i i?1 = f(n)10 =
1X
k=1
akf(n?1)k0 = f(n)00 ) fi i?1 = f00
x2d?ij x8fx64ixc4x67x14x88jx1bxdaxeaxa7x40x6f?i0 =?i i?1 +?i?1 0
xa7xcfx8f?i i?1;?i?1 0x83x70xd5xe1xa7xa4xb1
1X
n=1
P(?i0 = n)zn =
1X
k=1
P(?i i?1 = k)zk
1X
m=1
P(?i?1 0 = m)zm
) fi0 = fi i?1fi?1 0; (z = 1)
) fi0 = (f00)i
f00 =
1X
n=2
1X
k=1
P0kf(n?1)k0 +f(1)00 =
1X
k=1
P0kfk0 +a0 =
1X
k=0
ak(f00)k
11
x54x90xa7x330x141x53xc3x29xa7xa4xb1f00 = 1 ) 0x8fx7ex88x15x22
f(n)i i?1 = f(n)10 = f(n)00 )?i i?1 =?10 =?00
E(?i0) = E(?i i?1)+E(?i?1 0) )?i0 =?i i?1 +?i?1 0
)?i0 = i?00
10 =
1X
n=1
nf(n)10 =
1X
n=2
"
(n?1)
1X
k=1
P1kf(n?1)k0 +
1X
k=1
P1kf(n?1)k0
#
+P10
=
1X
k=1
P1k
1X
n=2
(n?1)f(n?1)k0 +
1X
k=1
P1k
1X
n=2
f(n?1)k0 +P10
=
1X
k=1
ak?k0 +
1X
k=1
P1kfk0 +a0
= 1+?10
1X
k=1
kak ; (fi0 = f00 = 1)
)?00 =?10 < 1 ; (
1X
k=1
kak < 1)
xa4xb10xb4x14x7ex88x15xa7x77x2c0xb4x9axb1xcfx1bxa7xbfx85x54xeaxbcxf3xb4xd8x8cx15xf3xa7xa4xb1x6bxb2xadxa9xd9x22
3.7 (1) x64… = …P x8cx1a… = (21=62;23=62;9=31)x22
x54xeaxbcxf3xb4xd8x8cx15x48x7bxf3xa7xa4xb1 limn!1p(n)ij = …jxa7x64x64x8cx1a
limn!1P(n) =
0
@
21
62
23
62
9
3121
62
23
62
9
3121
62
23
62
9
31
1
A
(2) x54xeaxbcxf3xb4xb2xadx4cxa7x1bxbfx87x5ex87xb4…(0)xb4xb2xadxa9xd9x22
EXn = 1:87;DXn = 0:77
3.8 (1)
p(T13 = n) = p(Xn = 3;Xk 6= 3;1? k? n?1;X0 = 1)
= p(Xn = 3;Xk 6= 3;1? k? n?1jX0 = 1)p(X0 = 1)
=
n?2X
k=0
pk11 p12 pn?k?222 p23 +pn?111 p13
=
‰ 1
4 n = 1
1
4
3
4
￠n?1 n? 2
ET13 = 4
(2)
f(1)11 = 1=2;f(1)22 = 3=4;f(1)33 = 1
f(n)11 = 0;f(n)22 = 0;f(n)33 = 0;n? 2
) f11 = 1=2;f22 = 3=4;f33 = 1
12
(3) xcfx8f1xda2xb4x9ax7ex88x1bxa7xa4xb1
limn!1p(n)31 = limn!1p(n)21 = limn!1p(n)32 = limn!1p(n)12 = limn!1p(n)11 = limn!1p(n)22 = 0
xcfx8fPnx7ax31x3x83x83xdax8f1xa7xa4xb1
limn!1Pn =
0
@
0 0 1
0 0 1
0 0 1
1
A
3.12 x79xb2xb5
1X
n=1
p(n)ij =
1X
n=1
nX
l=1
f(l)ij p(n?l)jj
=
1X
l=1
1X
n=l
f(l)ij p(n?l)jj
=
1X
l=1
f(l)ij
1X
n=l
p(n?l)jj
=
1X
l=1
f(l)ij
1X
n=0
p(n)jj
= fij1?f
jj
< 1;(jx9ax7ex88;fjj < 1)
3.13
P(Rn+1 = jjRn = in;￠￠￠;R1 = i1)
= P(Xn+1 = jjRn = in;￠￠￠;R1 = i1)
= P(Xn+1 = j) = P(Xn+1 = jjRn = in)
= aj;j > in
P(Rn+1 = injRn = in;￠￠￠;R1 = i1)
= P(Xn+1? injRn = in;￠￠￠;R1 = i1)
= P(Xn+1? in) = P(Xn+1? injRn = in)
=
inX
k=0
ak;j 6= in
P(Rn+1 = jjRn = in;￠￠￠;R1 = i1) = 0;j < in
xa4xb1fRi;i? 1gxb4xeaxbcxf3x22x3dxa3x56xc7x8fxb5
pij =
8>
>><
>>>:
aj j > i
iX
k=0
ak j = i
0 j < i
3.15 (1)xb7x82x79xb2x8dx98x84x1bx28xd8xb5pnk0 =
k
k+n?
k?1
k+n?1
1
bk;k = 0;1;￠￠￠xa7xd9xa5?1 = 0x22
xe6x5exeaxc6x38x42x7bx22
13
n = 1x9e
pk0 = b0(flk?flk+1) 1b
k
=
b
k
k?
bk+1
k+1
1
bk
=
k? k?1
k?
k+1? k
k+1
1
bk
=
k
k+1?
k?1
k
1
bk
x62x17n = m?1x9ex28xd8xa4xe1xa7x8n = mx9e
pmk0 =
kX
i=0
pkipm?1i0 +pk k+1pn?1k+1 0
=
kX
i=0
k
k+1?
k?1
k
b
i
bk
i
i+m?1?
i?1
i+m?2
1
bi +
bk+1 k
k+1bk
k+1
k+m?
k
k+m?1
1
bk+1
=
k
k+1?
k?1
k
k
k+m?1
1
bk +
k
k+1
k+1
k+m?
k
k+m?1
1
bk
=
k
k+m?
k?1
k+m?1
1
bk
x64x38x42x7bxa7x28xd8x1ax79x22
(2) x65x54xf3xb4x9ax7ex88xf3x4b0xb4x9ax7ex88x15)
1X
n=0
1
n =
1X
n=0
pn00 < 1x22
x58x4a
1X
n=0
1
n < 1x40x6f0xb4x9ax7ex88x15xa7x77x2c 0 $ k;k = 0;1;2;￠￠￠xa7x64x64x8cx1a k xb4x9ax7ex88
x15xa78k = 0;1;2;￠￠￠xa7xa4xb1x54xf3xb4x9ax7ex88xf3x22
3.17 x79xb2xb5
P(Xn+1jXk 2 Bk;0? k? n?1;Xn = i)
= P(Xn+1;Xk 2 Bk;0? k? n?1;Xn = i)P(X
k 2 Bk;0? k? n?1;Xn = i)
=
n?1X
k=0
X
lk2Bk
P(Xn+1;Xk = lk;0? k? n?1;Xn = i)
P(Xk 2 Bk;0? k? n?1;Xn = i)
=
n?1X
k=0
X
lk2Bk
P(Xn+1jXn = i)P(Xk = lk;0? k? n?1;Xn = i)
P(Xk 2 Bk;0? k? n?1;Xn = i) ;(xeaxbcx35)
=
P(Xn+1jXn = i)
n?1X
k=0
X
lk2Bk
P(Xk = lk;0? k? n?1;Xn = i)
P(Xk 2 Bk;0? k? n?1;Xn = i)
= P(Xn+1jXn = i)
3.20 (1) xf9xb4x98x87x9axb1xcfxd8x8cx15x1bx6bx81x47x15MCxa7x34x81xa9xd9x3dx8fxb2xadxa9xd9x22x64…P = …x1a… =
(44=81;1=3;10=81) xa7x40x6f
limn!1p(n)ij = …j =
8
<
:
44
81 j = 11
3 j = 210
81 j = 3
14
limn!1E(XnjX0 = 1) = limn!1
3X
j=1
jp(n)1j
=
3X
j=1
j limn!1p(n)1j = 128=81
(2)
fT1 = kg = fXi =2 S0;1? i? k?1;Xk 2 S0g
x3dx9dx36x75X1;X2;￠￠￠;Xkxa7x86Xk+1;Xk+2;￠￠￠xc3x27xa7xa4xb1T1xb4xcax9ex22
1x8fxb4x3dx9dx36x75X1;X2;￠￠￠;Xkxa7xdx86Xk+1;Xk+2;￠￠￠xc3x27xa7xa4xb1?1x8fxb4xcax9ex22
(3)
P(T1 = k) = P(Xi = 1;1? i? k?1;Xk 2 S0) = 38
5
8
k?1
ET1 =
1X
k=1
k 38
5
8
k?1
= 83
x2d
… = (P(XT1 = 2;P(XT1 = 3)=[P(XT1 = 2)+P(XT1 = 3)] = (2=3;1=3)
P =
1=2 1=6
1=4 0
E(?1) = ET1 +E(?1?T1) = 8=3+
1X
i=1
iP(?1?T1 = i)
= 8=3+
1X
i=1
i…Pi?1(1=3;3=4)T
= 8=3+…(I?P)?2(1=3;3=4)T
= 162=33
(4) xcfx8fTm? 2m?1xa4xb1N(3) = IT1?3 +IT2?3
P(N(3) = 0) = P(T1 > 3) = p311 = 125=512
P(N(3) = 1) = P(T1? 3;T2 > 3)
= P(T1 = 1;T2 > 3)+P(T1 = 2)+P(T1 = 3)
= 1?P(T1 = 1;T2 = 3)+P(T1 = 2)+P(T1 = 3)
= 1?(p12p21 +p13p31)(p12 +p13)+p11(p12 +p13)+p211(p12 +p13)
= 353=512
P(N(3) = 2) = P(T2? 3 < T3)
= P(T1 = 1;T2 = 3)
= (p12p21 +p13p31)(p12 +p13) = 34=512
P(N(3) = k) = 0;k? 3
P(N(4) = 2) = P(T2? 4 < T3)
= P(T1 = 1;T2 = 3)+P(T1 = 1;T2 = 4)+P(T1 = 2;T2 = 4)
= (p12p21 +p13p31)(p12 +p13)+(p12p22p21 +p12p23p31 +p13p33p31 +p13p32p21)
(p12 +p13)+p11(p12p21 +p13p31)(p12 +p13) = 0:1807
15
6.1
P00(t) = P(
1[
n=0
(N(t) = 2n)) =
1X
n=0
(?t)2n
(2n)! e
t = et +e?t
2 e
t = 1+e?2?t
2
P11(t) = P(
1[
n=0
(N(t) = 2n)) = 1+e
2?t
2
P01(t) = 1?P00(t) = 1?e
2?t
2
P10(t) = 1?P11(t) = 1?e
2?t
2
Q = P0(0) =
6.2 x646.3x21x7e1x1bx28xd8x8cx1axb5
E(X(t)) = P(X(t) = 1) = P1(t) =++?e?(?+?)t =
x64xbdx6e6.2.3x1axb5
E(?1jX(0) = 0) =
Z 1
0
P(?1 > tjX(0) = 0)dt =
Z 1
0
etdt = 1?
s > tx9exb5
cov(X(s);X(t)) = E(X(s)X(t))?E(X(s))E(X(t))
= P(X(s) = 1;X(t) = 1)?P(X(s) = 1)P(X(t) = 1)
= P11(s?t)P(X(t) = 1)?P(X(s) = 1)P(X(t) = 1)
=
+
+?e
(?+?)t
+?e
(?+?)s +?
+?e
(?+?)(s?t)
s < tx9exf2sx86tx2x86xa0x98x3dx8cx22
s = tx9exb5
cov(X(s);X(t)) = E(X2(t))?E(X(t))2
= P(X(t) = 1)?P(X(t) = 1)2
=
+
+?e
(?+?)t
+?e
(?+?)s +?
+?
E(X(s+t)jX(s) = 1) = P(X(s+t)jX(s) = 1) = P11(t) =++?e?(?+?)t
6.15 (1)
P(X(t) = ijX(t) 2 B) = P(X(t) = i;X(t) 2 B)P(X(t) 2 B)
= P(X(t) = i)P(X(t) 2 B) = P(X(t) = i)X
j2B
P(X(t) = j)
= PiX
j2B
Pj
(3)x17
i = infft,t > 0;X(0) = i;X(t) 6= ig;?0i = infft,t > 0;X(?i +t) 2 Gg
16
x40x6f
eFi(s) = E(e?sTijX(0) = i) = E(e?s(?i+?0i)jX(0) = i)
= E(e?s?ijX(0) = i)E(e?s?0ijX(0) = i) (?i;?0ix83x70xd5xe1)
=
Z 1
0
e?std(1?e?qit)E(E(e?s?0ijX(0) = ijX(?i)))
= qi(qi +s)?1E(E(e?s?0ijX(?i)))
= qi(qi +s)?1
X
j2S
E(e?s?0ijX(?i) = j)P(X(?i) = j)
= qi(qi +s)?1
2
4X
j2B
E(e?s?0ijX(?i) = j)Pij +
X
j2G
Pij
3
5
= qi(qi +s)?1
2
4X
j2B
eFj(s)Pij +X
j2G
Pij
3
5
(9)x58x4ax28xd8xa4xe1xa7x40x6f
E(e?sTvjX(t) 2 B) =
Z 1
0
e?suP(Tv > ujX(t?) 2 G;X(t) 2 B)du
￡fE(TvjX(t?) 2 G;X(t) 2 B)g?1
=
1
s?
1
s
Z 1
0
e?suf(Tv > ujX(t?) 2 G;X(t) 2 B)du
￡fE(TvjX(t?) 2 G;X(t) 2 B)g?1
= ￡1?E(e?sTvjX(t?) 2 G;X(t) 2 B)?
￡fsE(TvjX(t?) 2 G;X(t) 2 B)g?1
x64x2excax2ex64x43x86x1bx8dx98x35x8cx1axb5
P(Tv? sjX(t) 2 B) =
Z s
0
P(Tv > ujX(t?) 2 G;X(t) 2 B)du
￡fsE(TvjX(t?) 2 G;X(t) 2 B)g?1
6.21
ePij = P(X(?n+1) = jjX(?n = i)
= P(X(?1) = jjX(0) = i)
=
‰ qij
qi j 6= i
0 j = i
eP =
0
@
0 3=5 2=5
1=4 0 3=4
1=3 2=3 0
1
A
P(N(t) = 1) = P(?1? t;?2 > t)
=
2X
i;j=0
Z t
0
P(?2?u > t?ujX(?1) = j;?1 = u;X(0) = i)
dP(?1? u;X(?1) = jjX(0) = i)P(X(0) = i)
= 23
Z t
0
e?5(t?u)d
1
4(1?e
4u)
+ 23
Z t
0
e?6(t?u)d
3
4(1?e
4u)
17
+13
Z t
0
e?5(t?u)d
1
3(1?e
6u)
+ 13
Z t
0
e?4(t?u)d
2
3(1?e
6u)
= 23e?5t(et?1)+e?6t(e2t?1)+ 23e?5t(1?e?t)+ 23e?4t(1?e?2t)
= 53e?4t? 43e?6t? 13
(2) x64xbdx6e6.10.3x1a
Φ1(s) = (s+q1)?1q10 +(s+q1)?1q12Φ2(s)
Φ2(s) = (s+q2)?1q20 +(s+q2)?1q21Φ1(s)
x64x64x8cx1a
Φ1(s) = s+12s2 +10s+12
Φ2(s) = s+8s2 +10s+12
xcfx8fP(T1 < t) = 23P(T1 < tjX(0) = 1)+ 13P(T1 < tjX(0) = 2)xa7xa4xb1P(T1 < t)x1bLaplacex43x86
x8fxb5
Φ(s) = 23Φ1(s)+ 13Φ2(s) = s+32=3s2 +10s+12
x64x64x8cx1axb5
E(T1) =?Φ0(0) = 71108
E(T21) = Φ00(0) = 307324
E(T31) =?Φ000(0) = 661324
E(T41) = Φ0000(0) = 5689972
(3) x17
= infft,t > 0;X(t) 6= X(0)g
E(T2) = E(T2?T1)+E(T1) = E(?jX(0) = 0)+E(T1)
= 1q
0
+ 71108 = 463540
P(X(T2) = 2) = P(X(?) = 2jX(0) = 0) = q02q
0
= 25
18
x56xc7x1bx8cx5cx35x8cx1ax22
(4) xcfx8fB ‰ Axa7xa4xb1A = BSA?Bxa7x64x64x8cx1ax22
1.3 x8ATB 6= `x9exa7 (A)x1x6b16x87x3x83x22x8ATB = `x9exa7 (A)x6b8x87x3x83x22xd8x4ax15xd1
1.4 x4bx38xa5A3 = f(a;b),a;b 2 Rg
(1) xcfx8f8n;A ‰ Anxa4xb1x6bA ‰
1\
n=1
Anx22
x65xa1x79xb2
1\
n=1
An ‰ Ax22x87x79x7bxb5x62x179x 2
1\
n=1
An;x =2 Axa7x40x6fx? a > 0xa7x12n = d 1x?ae + 1xa7
x4bx > a + 1nxa7x64x64x8cx1ax =2 Anxa7x67xf1x22xa4xb1
1\
n=1
An ‰ Ax22x64x64x8cx1aA =
1\
n=1
Anx22x61x71x8c
x1aB =
1[
n=1
Bnx22
(2) xcfx8f8a 2 R;(?1;a] =
1[
n=1
(?n;a]xa7xa4xb1A1 ‰ (A2)x22xcfx8f8a;b 2 R;(a;b] = (?1;b]?
(?1;a]xa7xa4xb1A2 ‰ (A1)x22x6exfex8cx1a (A1) = (A2)
(3)
8a 2 R;(?1;a] =
1[
n=1
(?n;a]
=
1\
m=1
(?1;a+ 1m)
=
1\
m=1
1[
n=1
(?n;a+ 1m)
=
1\
m=1
1[
n=1
[?n;a+ 1m)
=
1[
n=1
[?n;a]
8a;b 2 R;(a;b] = (?1;b]?(?1;a]
(a;b) =
1[
n=1
(?1;b? 1n]?(?1;a]
[a;b) =
1[
n=1
(?1;b? 1n]?
1[
m=1
(?1;a? 1m]
[a;b] = (?1;b]?
1[
m=1
(?1;a? 1m]
x64x64x8cx1a (A1) = (Ai);2? i? 5
1.6 (3) xcfx8fA2k+1 ‰ A2kxa7xa4xb1
1[
k=n
Ak =
‰ S1
k=n=2 A2k = (
1
4;
1
2 +
1
n] nx8fxf3xeaS1
k=(n+1)=2 A2k = (
1
4;
1
2 +
1
n+1] nx8fxdbxea
1
)
1\
n=1
1[
k=n
Ak =
1\
n=1
(14; 12 + 1n] = (14; 12]
x64x64x8cx1aP
1\
n=1
1[
k=n
Ak
!
= F(12)?F(14)x22
1\
k=n
Ak = ` )
1[
n=1
1\
k=n
Ak = `
x64x64x8cx1aP
1[
n=1
1\
k=n
Ak
!
= 0x22
1.10 (1) x64x34x81x1bxbdxc2
! 2f!,limn!1Xn = Xg
,8m? 1 9n? 1 8k? n jXk(!)?Xj < 1=m
,8m? 1 9n? 1 ! 2
\
k?n
f!,jXk?Xj < 1=mg
,8m? 1 ! 2
[
n?1
\
k?n
f!,jXk?Xj < 1=mg
,! 2
\
m?1
[
n?1
\
k?n
f!,jXk?Xj < 1=mg
1.14
EN =
1X
n=1
nP(N = n) =
1X
n=1
nX
m=1
P(N = n)
=
1X
n=1
1X
m=n
P(N = m) =
1X
n=1
P(N > n)
EX =
Z 1
0
xdF(x) =
Z 1
0
(
Z x
0
dy)dF(x)
=
Z 1
0
Z 1
y
dF(x)dy =
Z 1
0
(1?F(x))dx
E(Xn) =
Z 1
0
xndF(x) =
Z 1
0
(
Z x
0
nyn?1dy)dF(x)
=
Z 1
0
Z 1
y
dF(x)nyn?1dy =
Z 1
0
nxn?1(1?F(x))dx
1.18
P(? = k) =
1X
n=k
P(N = n)P(? = kjN = n)
=
1X
n=k
P(N = n)P(
nX
i=1
Xi = kjN = n)
=
1X
n=k
P(N = n)P(
nX
i=1
Xi = k)
2
=
1X
n=k
ne
n! C
k
np
k(1?p)n?k
=
1X
n=k
ne
k!(n?k)!p
k(1?p)n?k
= e
kpk
k!
1X
n=k
n?k
(n?k)!(1?p)
n?k
=
1X
n=k
ne
k!(n?k)!p
k(1?p)n?k
= e
kpk
k! e
(1?p)
= (?p)
k
k! e
p? Po(?p)
xa4xb1E(?) =?p; D(?) =?px22
1.19 (1)
P(N1 +N2 = n) =
nX
k=0
P(N1 +N2 = n;N1 = k)
=
nX
k=0
P(N2 = n?k)P(N1 = k)
=
nX
k=0
k1e1
k!
n?k2 e2
(n?k)!
= (?1 +?2)
n
n! e
(?1+?2)
(2)
P(N1 = kjN1 +N2 = n) = P(N1 = k)P(N2 = n?k)P(N
1 +N2 = n
= Ckn
1
1 +?2
k?
2
1 +?2
n?k
(3)
P(N1 +N2jN3) = P(N1jN3)+P(N2jN3) = P(N1)+P(N2) = P(N1 +N2)
xa4xb1N1 +N2x86N3xd5xe1x22
(4)
E(N1jN1 +N2 = n) =
nX
k=0
kP(N1 = kjN1 +N2 = n)
=
nX
k=0
kCkn
1
1 +?2
k?
2
1 +?2
n?k
= n?1?
1 +?2
nX
k=1
Ck?1n?1
1
1 +?2
k?1?
2
1 +?2
n?k
= n?1?
1 +?2
3
xa4xb1E(N1jN1 +N2) =?1?
1 +?2
(N1 +N2)x22
E(N1 +N2jN1) = E(N1jN1)+E(N2jN1) = N1 +E(N2) = N1 +?2
1.20 xf9x70x90x89xd1x31x6ex2x4bx1bx79xb2xa7x63xfcx87x61x71x8cx1ax22
(3)
E(E(IAjIB)jIB;IC) = E(P(AjB)IB +P(AjB)IBjIB;IC)
= E(P(AjB)IBjIB;IC)+E(P(AjB)IBjIB;IC)
= P(AjB)IB +P(AjB)IB
= E(IAjIB)
1.24 X;Yx1bxe9xdcx56xc7x97xddxbcxeax8fxb5
f(X;Y)(x;y) = e?(x+y)I(x>0;y>0)
x64 X = UV=(1+V);Y = U=(1+V) x1aJacobixddxax1bx31xfxaax8fxb5
J =?u=(1+v)2
x64x64x8cx1axb5
f(U;V)(u;v) = e?u u(1+v)2I(u > 0;v > 0)
xa9x4fxc8xa9x8cx1ax3exexa9xd9x8fxb5
fU(u) = ue?uI(u > 0)
fV (v) = 1(1+v)2I(v > 0)
1.25
P(Z? z) = P(X +Y? z)
=
nX
k=0
P(X +Y? z;X = k)
=
nX
k=0
P(Y? z?k)P(X = k)
x64x64x8cx1aZx1bxa9xd9xbcxeaxa7x32xe9xa9xd9xbcxeaxa6x13x8cx1ax56xc7x97xddxbcxeax22
1.28
E(E(XjY;Z)jY)
=
X
i
E(
X
j;k
E(XjY = yj;Z = zk)I(Y=yj)I(Z=zk)jY = yi)I(Y=yi)
=
X
j;k
E(XjY = yj;Z = zk)E(I(Z=zk)jY = yj)I(Y=yj)
xcfx8fx8i 6= jx9eE(I(Y=yj)I(Z=zk)jY = yj) = 0
=
X
j;k
X
i
xiP(X = xijY = yj;Z = zk)P(Z = zkjY = yj)I(Y=yj)
4
=
X
j;k
X
i
xiP(X = xi;Z = zkjY = yj)I(Y=yj)
=
X
j
X
i
xiP(X = xijY = yj)I(Y=yj)
=
X
j
E(XjY = yj)I(Y=yj)
= E(XjY)
x2cx98x87x61x71x8cx1ax22
2.2 x79xb2xb5
P(N(s) = kjN(t) = n)
= P(N(s) = k;N(t) = n)P(N(t) = n)
= P(N(s) = k;N(t)?N(s) = n?k)P(N(t) = n)
= P(N(s) = k)P(N(t?s) = n?k)P(N(t) = n)
=
(?s)k
k! e
s(?(t?s))n?k
(n?k)! e
(t?s)
(?t)k
k! e
t
= Ckn(st)k(1? st)n?k (0? k? n)
2.3 x29xb5(1)
E(N(t)N(s+t))
= E(N(t)2 +N(t)(N(s+t)?N(t)))
= E(N(t)2)+E(N(t))E(N(s))
= (?t)2 +?t+?2ts
(2)
E(N(s+t)jN(s))
= E(N(s))+E(N(s+t)?N(s)jN(s))
= N(s)+E(N(t)jN(s))
= N(s)+?t
P(E(N(s+t)jN(s)) = n+?t) = P(N(s) = n) = (?s)
n
n! e
s
2.5 x29xb5x17Ui? U[0;t]; i = 1;2;￠￠￠;nxa7x4bxd9x5ex53xdax4fxfex86S1;S2;￠￠￠;Snx33
N(t) = n x1bx5ex87x65x1bxa9xd9x83xd3x22
E(SkjN(t) = n) = E(U(k)) = ktn+1; (k? n)
5
2.6 x29xb5x8x? 0x9e
P(W(t)? x) = P(N(t+x)?N(t) > 0)
= 1?P(N(t+x)?N(t) = 0)
= 1?ex
x8x < 0x9exa7P(W(t)? x) = 0x22
xa4xb1P(W(t)? x) = 1?e?xIx?0x22
x80 < x < tx9e
P(V(t)? x) = P(N(t)?N(t?x) > 0)
= 1?P(N(t)?N(t?x) = 0)
= 1?ex
x8x? tx9exa7P(V(t)? x) = 1x22
x8x? 0x9exa7P(V(t)? x) = 0x22
xa4xb1
P(V(t)? x)
8<
:
1?e?x ; 0 < x < t
1 ; x? t
0 ; x? 0
x8x > 0;0 < y < tx9e
P(W(t)? x;V(t)? y)
= P(N(t+x)?N(t) > 0;N(t)?N(t?y) > 0)
= P(N(t+x)?N(t) > 0)P(N(t)?N(t?y) > 0)
= (1?ex)(1?ey)
x8y? tx9e
P(W(t)? x;V(t)? y)
= P(N(t+x)?N(t) > 0)
= P(N(t+x)?N(t) > 0)
= 1?ex
2.10 x29xb5x64x4bxbfx7f? = 1=2xa7x40x6f
P(Sn > x) =
Z 1
x
(?t)n?1
(n?1)! e
tdt
= 12n(n?1)!
Z 1
x
tn?1e?t=2dt
2.15 x29xb5x64xc4x1dx8dx23xbdx6e
limx!1 E(N(x))x = 1E(X +Y) = 23
2.16 x29xb5x2dYi = Xi?– xa7x4bYi? Ex(‰)xa7xa4xb1x64Yix29xa4x1bxb4xd1x74x36x22
P(N(t)? n)
6
= P(Sn? t)
= P(
nX
i=1
Xi? t)
= P(
nX
i=1
Yi? t?n–)
= 1?
n?1X
k=0
[?(t?n–)]k
k! e
(t?n–)
2.17 x29xb5
F(s) =
Z 1
0
e?st?2tetdt
=?
2
(s+?)2
)?m(s) =
F(s)
1F(s)
=?
2
s(s+2?)
) m0(t) =?22e?2?t
) m(t) = (?2t+ 14e?2?t? 14)I(t?0)
2.18 x29xb5x80 < x < tx9e
P(flt? x) = P(W(t)+V(t)? x)
=
Z x
0
P(W(t)? x?yjV(t) = y)dP(V(t)? y)
=
Z x
0
P(W(t)? x?y)dP(V(t)? y)
=
Z x
0
(1?e(x?y))eydy
= 1?exxex
x8x? tx9e
P(flt? x) = P(W(t)+V(t)? x)
=
Z x
0
P(W(t)? x?yjV(t) = y)dP(V(t)? y)
=
Z x
0
P(W(t)? x?y)dP(V(t)? y)
=
Z t?
0
(1?e(x?y))eydy +
Z 1
t?
P(W(t)? x?yjV(t) = y)dP(V(t)? y)
= 1?ettet +(1?e(x?t))et
= 1?extex
2.25 x29xb5
7
(1) S1;S2?S1;￠￠￠;Sn?Sn?1xd3xa9xd9x2xd8xb4x5ex87xd5xe1x22
(2) x8n = 0x9e
E(S1jN(t) = 0) = E(W(t)+tjN(t) = 0)
= t+E(W(t))
= t+ 1?
x8n? 1x9e
E(S1jN(t) = n) = E(U(1))
= tn+1
(3) x8n? kx9e
E(SkjN(t) = n) = E(xk +xk?1 +￠￠￠+xn+2 +W(t)+tjN(t) = n)
= t+ k?n?
x8n? kx9e
E(SkjN(t) = n) = E(U(k))
= ktn+1
(4) x86U(i);U(k)x1bxe9xdcxa9xd9x83xd3xa7x8cx5ex87x3x7bxbdxf6xc8xa9x1ax14x22
2.26 x29xb5
(1)
f(x;y) = lim
h!0
P(x < S2? x+h;y < S5? y +h
h2
= lim
h!0
P(N(x) = 1;N(x+h)?N(x) = 1;N(y)?N(x+h) = 2;N(y +h)?N(y = 1))
h2
= lim
h!0
xex?heh?2(y?x?h)22 e(y?x?h)?heh
h2
=?
5x(y?x)2ey
2
(2)
E(S1) = E(S1jN(t)? 1)P(N(t)? 1)+E(S1jN(t) = 0)P(N(t) = 0)
) E(S1jN(t)? 1)) = E(S1)?E(S1jN(t) = 0)P(N(t) = 0)P(N(t)? 1)
E(S1jN(t)? 1)) = E(S1)?E(S1jN(t) = 0)P(N(t) = 0)P(N(t)? 1)
=
1
(t+
1
)e
t
1?et
8
(3)
f(x;y) = lim
h!0
P(x < S1? x+h;y < S2? y +h;N(t) = 1
h2
= lim
h!0
P(N(x) = 0;N(x+h)?N(x) = 1;
N(t)?N(x+h) = 0;N(y)?N(t) = 0;N(y +h)?N(y = 1))=h2
= lim
h!0
ex?hehe(t?x?h)e(y?h?t)?heh
h2
=?t e(y?t)
2.29 x29xb5
(1) x8n = 0x9exa7
x65x? t; P(S1? x;N(t) = 0) = 0x22
x65x > t
P(S1? x;N(t) = 0)
= P(S1? xjN(t) = 0)P(N(t) = 0)
= P(W(t)? x?tjN(t) = 0)P(N(t) = 0)
= P(W(t)? x?t)P(N(t) = 0)
= (1?e(x?t))et
= et?ex
x8n? 1x9e
x65x < 0; P(S1? x;N(t) = n) = 0x22
x65x? 0
P(S1? x;N(t) = n)
= P(S1? xjN(t) = n)P(N(t) = n)
= P(U(1)? x)P(N(t) = n)I(x?t) +P(N(t) = n)I(x>t)
= [1?
t?x
t
n
](?t)
net
n! I(x?t) +
(?t)net
n! I(x>t)
(2) x8n = 0x9exa7
x65x? t; P(S2? x;N(t) = 0) = 0x22
x65x > t
P(S2? x;N(t) = 0)
= P(N(t) = 0;N(x)?N(t)? 2)
= P(N(t) = 0)P(N(x)?N(t)? 2)
= et?ex(x?t)ex
x8n = 1x9e
x65x? t; P(S2? x;N(t) = 1) = 0x22
x65x > t
P(S2? x;N(t) = 1)
= P(S2? xjN(t) = 1)P(N(t) = 1)
= P(W(t)? x?tjN(t) = 1)P(N(t) = 1)
= P(W(t)? x?t)P(N(t) = 1)
= (1?e(x?t))?tet
9
x8n? 2x9e
x65x? 0; P(S2? x;N(t) = n) = 0x22
x65x > 0
P(S2? x;N(t) = n)
= P(S2? xjN(t) = n)P(N(t) = n)
= P(U(2)? x)P(N(t) = n)I(x?t) +P(N(t) = n)I(x>t)
=
"
1?n
t?x
t
n?1 x
t?
t?x
t
n# (?t)net
n! I(x?t) +
(?t)net
n! I(x>t)
= ￡tn?n(t?x)n?1x?(t?x)n? (?t)
net
n! I(x?t) +
(?t)net
n! I(x>t)
3.1 (1)
E(X2) = 1￡ 19 +2￡ 29 +3￡ 23 = 239
E(X2jX1 = i) =
3X
j=1
j P(X2 = jjX1 = i) =
3X
j=1
j Pij
)
8<
:
E(X2jX1 = 1) = 1
E(X2jX1 = 2) = 73
E(X2jX1 = 3) = 83
) E(X2jX1) = I(X1=1) + 73I(X1=2) + 83I(X1=3)
xd3x6eE(X3jX2) = I(X2=1) + 73I(X2=2) + 83I(X2=3)
(2)
P(T = 1jX0 = 3) = P(X1 = 1jX0 = 3) = P31 = 0
P(T = 2jX0 = 3) = P(X2 = 1;X1 6= 1jX0 = 3) =
3X
i=2
P3iPi1 = 19
P(T = 3jX0 = 3) = P(X3 = 1;X1 6= 1;X2 6= 1jX0 = 3)
=
3X
i=2
3X
j=2
P(X3 = 1;X2 = i;X1 = jjX0 = 3)
=
3X
i=2
3X
j=2
P3jPjiPi1 = 227
P(T? 4jX0 = 3) = 1?
3X
k=1
P(T = kjX0 = 3) = 2227
E(T ^4jX0 = 3) =
3X
i=1
i P(T = ijX0 = 3)+4P(T? 4jX0 = 3)
= 10027
(3)
P(T11 = 1jX0 = 1) = P(X1 = 1jX0 = 1) = 0
P(T11 = 2kjX0 = 1) = P12(P23P32)k?1P21 +P13(P32P23)k?1P31
10
= 13
4
9
k?1
P(T11 = 2k +1jX0 = 1) = P12(P23P32)k?1P23P31 +P13(P32P23)k?1P32P21
= 29
4
9
k?1
) ET11 =
1X
k=1
2k13
4
9
k?1
+
1X
k=1
(2k +1)29
4
9
k?1
= 4
3.2 x79xb2xb5x5exeaxc6x38x42x7bx22
n = 1x9exa7xb7x4bx77x2cxa4xe1x22
x62x17n = kx9exb7x4bxa4xe1x22
x40x6fx8n = k +1x9exa7
Pk+1 = Pk P
= 1a+b
b a
b a
1?a a
b 1?b
+
(1?a?b)k
a+b
a?a
b b
1?a a
b 1?b
= 1a+b
b a
b a
+ (1?a?b)
k
a+b (1?a?b)
a?a
b b
= 1a+b
b a
b a
+ (1?a?b)
k+1
a+b
a?a
b b
x64xeaxc6x38x42x7bxb7x4bx1ax79x22
3.3 x79xb2xb5
`ij(z) =
1X
n=0
P(n)ij zn =
1X
n=0
P nijzn
) `(z) =
1X
n=0
zPn = (I?zP)?1
xd9xa5Pnij xb4x90xaPnx1bx31ix31x31jxfx1bx3x83x22
3.6
f(n)i i?1 = f(n)10 =
1X
k=1
akf(n?1)k0 = f(n)00 ) fi i?1 = f00
x2d?ij x8fx64ixc4x67x14x88jx1bxdaxeaxa7x40x6f?i0 =?i i?1 +?i?1 0
xa7xcfx8f?i i?1;?i?1 0x83x70xd5xe1xa7xa4xb1
1X
n=1
P(?i0 = n)zn =
1X
k=1
P(?i i?1 = k)zk
1X
m=1
P(?i?1 0 = m)zm
) fi0 = fi i?1fi?1 0; (z = 1)
) fi0 = (f00)i
f00 =
1X
n=2
1X
k=1
P0kf(n?1)k0 +f(1)00 =
1X
k=1
P0kfk0 +a0 =
1X
k=0
ak(f00)k
11
x54x90xa7x330x141x53xc3x29xa7xa4xb1f00 = 1 ) 0x8fx7ex88x15x22
f(n)i i?1 = f(n)10 = f(n)00 )?i i?1 =?10 =?00
E(?i0) = E(?i i?1)+E(?i?1 0) )?i0 =?i i?1 +?i?1 0
)?i0 = i?00
10 =
1X
n=1
nf(n)10 =
1X
n=2
"
(n?1)
1X
k=1
P1kf(n?1)k0 +
1X
k=1
P1kf(n?1)k0
#
+P10
=
1X
k=1
P1k
1X
n=2
(n?1)f(n?1)k0 +
1X
k=1
P1k
1X
n=2
f(n?1)k0 +P10
=
1X
k=1
ak?k0 +
1X
k=1
P1kfk0 +a0
= 1+?10
1X
k=1
kak ; (fi0 = f00 = 1)
)?00 =?10 < 1 ; (
1X
k=1
kak < 1)
xa4xb10xb4x14x7ex88x15xa7x77x2c0xb4x9axb1xcfx1bxa7xbfx85x54xeaxbcxf3xb4xd8x8cx15xf3xa7xa4xb1x6bxb2xadxa9xd9x22
3.7 (1) x64… = …P x8cx1a… = (21=62;23=62;9=31)x22
x54xeaxbcxf3xb4xd8x8cx15x48x7bxf3xa7xa4xb1 limn!1p(n)ij = …jxa7x64x64x8cx1a
limn!1P(n) =
0
@
21
62
23
62
9
3121
62
23
62
9
3121
62
23
62
9
31
1
A
(2) x54xeaxbcxf3xb4xb2xadx4cxa7x1bxbfx87x5ex87xb4…(0)xb4xb2xadxa9xd9x22
EXn = 1:87;DXn = 0:77
3.8 (1)
p(T13 = n) = p(Xn = 3;Xk 6= 3;1? k? n?1;X0 = 1)
= p(Xn = 3;Xk 6= 3;1? k? n?1jX0 = 1)p(X0 = 1)
=
n?2X
k=0
pk11 p12 pn?k?222 p23 +pn?111 p13
=
‰ 1
4 n = 1
1
4
3
4
￠n?1 n? 2
ET13 = 4
(2)
f(1)11 = 1=2;f(1)22 = 3=4;f(1)33 = 1
f(n)11 = 0;f(n)22 = 0;f(n)33 = 0;n? 2
) f11 = 1=2;f22 = 3=4;f33 = 1
12
(3) xcfx8f1xda2xb4x9ax7ex88x1bxa7xa4xb1
limn!1p(n)31 = limn!1p(n)21 = limn!1p(n)32 = limn!1p(n)12 = limn!1p(n)11 = limn!1p(n)22 = 0
xcfx8fPnx7ax31x3x83x83xdax8f1xa7xa4xb1
limn!1Pn =
0
@
0 0 1
0 0 1
0 0 1
1
A
3.12 x79xb2xb5
1X
n=1
p(n)ij =
1X
n=1
nX
l=1
f(l)ij p(n?l)jj
=
1X
l=1
1X
n=l
f(l)ij p(n?l)jj
=
1X
l=1
f(l)ij
1X
n=l
p(n?l)jj
=
1X
l=1
f(l)ij
1X
n=0
p(n)jj
= fij1?f
jj
< 1;(jx9ax7ex88;fjj < 1)
3.13
P(Rn+1 = jjRn = in;￠￠￠;R1 = i1)
= P(Xn+1 = jjRn = in;￠￠￠;R1 = i1)
= P(Xn+1 = j) = P(Xn+1 = jjRn = in)
= aj;j > in
P(Rn+1 = injRn = in;￠￠￠;R1 = i1)
= P(Xn+1? injRn = in;￠￠￠;R1 = i1)
= P(Xn+1? in) = P(Xn+1? injRn = in)
=
inX
k=0
ak;j 6= in
P(Rn+1 = jjRn = in;￠￠￠;R1 = i1) = 0;j < in
xa4xb1fRi;i? 1gxb4xeaxbcxf3x22x3dxa3x56xc7x8fxb5
pij =
8>
>><
>>>:
aj j > i
iX
k=0
ak j = i
0 j < i
3.15 (1)xb7x82x79xb2x8dx98x84x1bx28xd8xb5pnk0 =
k
k+n?
k?1
k+n?1
1
bk;k = 0;1;￠￠￠xa7xd9xa5?1 = 0x22
xe6x5exeaxc6x38x42x7bx22
13
n = 1x9e
pk0 = b0(flk?flk+1) 1b
k
=
b
k
k?
bk+1
k+1
1
bk
=
k? k?1
k?
k+1? k
k+1
1
bk
=
k
k+1?
k?1
k
1
bk
x62x17n = m?1x9ex28xd8xa4xe1xa7x8n = mx9e
pmk0 =
kX
i=0
pkipm?1i0 +pk k+1pn?1k+1 0
=
kX
i=0
k
k+1?
k?1
k
b
i
bk
i
i+m?1?
i?1
i+m?2
1
bi +
bk+1 k
k+1bk
k+1
k+m?
k
k+m?1
1
bk+1
=
k
k+1?
k?1
k
k
k+m?1
1
bk +
k
k+1
k+1
k+m?
k
k+m?1
1
bk
=
k
k+m?
k?1
k+m?1
1
bk
x64x38x42x7bxa7x28xd8x1ax79x22
(2) x65x54xf3xb4x9ax7ex88xf3x4b0xb4x9ax7ex88x15)
1X
n=0
1
n =
1X
n=0
pn00 < 1x22
x58x4a
1X
n=0
1
n < 1x40x6f0xb4x9ax7ex88x15xa7x77x2c 0 $ k;k = 0;1;2;￠￠￠xa7x64x64x8cx1a k xb4x9ax7ex88
x15xa78k = 0;1;2;￠￠￠xa7xa4xb1x54xf3xb4x9ax7ex88xf3x22
3.17 x79xb2xb5
P(Xn+1jXk 2 Bk;0? k? n?1;Xn = i)
= P(Xn+1;Xk 2 Bk;0? k? n?1;Xn = i)P(X
k 2 Bk;0? k? n?1;Xn = i)
=
n?1X
k=0
X
lk2Bk
P(Xn+1;Xk = lk;0? k? n?1;Xn = i)
P(Xk 2 Bk;0? k? n?1;Xn = i)
=
n?1X
k=0
X
lk2Bk
P(Xn+1jXn = i)P(Xk = lk;0? k? n?1;Xn = i)
P(Xk 2 Bk;0? k? n?1;Xn = i) ;(xeaxbcx35)
=
P(Xn+1jXn = i)
n?1X
k=0
X
lk2Bk
P(Xk = lk;0? k? n?1;Xn = i)
P(Xk 2 Bk;0? k? n?1;Xn = i)
= P(Xn+1jXn = i)
3.20 (1) xf9xb4x98x87x9axb1xcfxd8x8cx15x1bx6bx81x47x15MCxa7x34x81xa9xd9x3dx8fxb2xadxa9xd9x22x64…P = …x1a… =
(44=81;1=3;10=81) xa7x40x6f
limn!1p(n)ij = …j =
8
<
:
44
81 j = 11
3 j = 210
81 j = 3
14
limn!1E(XnjX0 = 1) = limn!1
3X
j=1
jp(n)1j
=
3X
j=1
j limn!1p(n)1j = 128=81
(2)
fT1 = kg = fXi =2 S0;1? i? k?1;Xk 2 S0g
x3dx9dx36x75X1;X2;￠￠￠;Xkxa7x86Xk+1;Xk+2;￠￠￠xc3x27xa7xa4xb1T1xb4xcax9ex22
1x8fxb4x3dx9dx36x75X1;X2;￠￠￠;Xkxa7xdx86Xk+1;Xk+2;￠￠￠xc3x27xa7xa4xb1?1x8fxb4xcax9ex22
(3)
P(T1 = k) = P(Xi = 1;1? i? k?1;Xk 2 S0) = 38
5
8
k?1
ET1 =
1X
k=1
k 38
5
8
k?1
= 83
x2d
… = (P(XT1 = 2;P(XT1 = 3)=[P(XT1 = 2)+P(XT1 = 3)] = (2=3;1=3)
P =
1=2 1=6
1=4 0
E(?1) = ET1 +E(?1?T1) = 8=3+
1X
i=1
iP(?1?T1 = i)
= 8=3+
1X
i=1
i…Pi?1(1=3;3=4)T
= 8=3+…(I?P)?2(1=3;3=4)T
= 162=33
(4) xcfx8fTm? 2m?1xa4xb1N(3) = IT1?3 +IT2?3
P(N(3) = 0) = P(T1 > 3) = p311 = 125=512
P(N(3) = 1) = P(T1? 3;T2 > 3)
= P(T1 = 1;T2 > 3)+P(T1 = 2)+P(T1 = 3)
= 1?P(T1 = 1;T2 = 3)+P(T1 = 2)+P(T1 = 3)
= 1?(p12p21 +p13p31)(p12 +p13)+p11(p12 +p13)+p211(p12 +p13)
= 353=512
P(N(3) = 2) = P(T2? 3 < T3)
= P(T1 = 1;T2 = 3)
= (p12p21 +p13p31)(p12 +p13) = 34=512
P(N(3) = k) = 0;k? 3
P(N(4) = 2) = P(T2? 4 < T3)
= P(T1 = 1;T2 = 3)+P(T1 = 1;T2 = 4)+P(T1 = 2;T2 = 4)
= (p12p21 +p13p31)(p12 +p13)+(p12p22p21 +p12p23p31 +p13p33p31 +p13p32p21)
(p12 +p13)+p11(p12p21 +p13p31)(p12 +p13) = 0:1807
15
6.1
P00(t) = P(
1[
n=0
(N(t) = 2n)) =
1X
n=0
(?t)2n
(2n)! e
t = et +e?t
2 e
t = 1+e?2?t
2
P11(t) = P(
1[
n=0
(N(t) = 2n)) = 1+e
2?t
2
P01(t) = 1?P00(t) = 1?e
2?t
2
P10(t) = 1?P11(t) = 1?e
2?t
2
Q = P0(0) =
6.2 x646.3x21x7e1x1bx28xd8x8cx1axb5
E(X(t)) = P(X(t) = 1) = P1(t) =++?e?(?+?)t =
x64xbdx6e6.2.3x1axb5
E(?1jX(0) = 0) =
Z 1
0
P(?1 > tjX(0) = 0)dt =
Z 1
0
etdt = 1?
s > tx9exb5
cov(X(s);X(t)) = E(X(s)X(t))?E(X(s))E(X(t))
= P(X(s) = 1;X(t) = 1)?P(X(s) = 1)P(X(t) = 1)
= P11(s?t)P(X(t) = 1)?P(X(s) = 1)P(X(t) = 1)
=
+
+?e
(?+?)t
+?e
(?+?)s +?
+?e
(?+?)(s?t)
s < tx9exf2sx86tx2x86xa0x98x3dx8cx22
s = tx9exb5
cov(X(s);X(t)) = E(X2(t))?E(X(t))2
= P(X(t) = 1)?P(X(t) = 1)2
=
+
+?e
(?+?)t
+?e
(?+?)s +?
+?
E(X(s+t)jX(s) = 1) = P(X(s+t)jX(s) = 1) = P11(t) =++?e?(?+?)t
6.15 (1)
P(X(t) = ijX(t) 2 B) = P(X(t) = i;X(t) 2 B)P(X(t) 2 B)
= P(X(t) = i)P(X(t) 2 B) = P(X(t) = i)X
j2B
P(X(t) = j)
= PiX
j2B
Pj
(3)x17
i = infft,t > 0;X(0) = i;X(t) 6= ig;?0i = infft,t > 0;X(?i +t) 2 Gg
16
x40x6f
eFi(s) = E(e?sTijX(0) = i) = E(e?s(?i+?0i)jX(0) = i)
= E(e?s?ijX(0) = i)E(e?s?0ijX(0) = i) (?i;?0ix83x70xd5xe1)
=
Z 1
0
e?std(1?e?qit)E(E(e?s?0ijX(0) = ijX(?i)))
= qi(qi +s)?1E(E(e?s?0ijX(?i)))
= qi(qi +s)?1
X
j2S
E(e?s?0ijX(?i) = j)P(X(?i) = j)
= qi(qi +s)?1
2
4X
j2B
E(e?s?0ijX(?i) = j)Pij +
X
j2G
Pij
3
5
= qi(qi +s)?1
2
4X
j2B
eFj(s)Pij +X
j2G
Pij
3
5
(9)x58x4ax28xd8xa4xe1xa7x40x6f
E(e?sTvjX(t) 2 B) =
Z 1
0
e?suP(Tv > ujX(t?) 2 G;X(t) 2 B)du
￡fE(TvjX(t?) 2 G;X(t) 2 B)g?1
=
1
s?
1
s
Z 1
0
e?suf(Tv > ujX(t?) 2 G;X(t) 2 B)du
￡fE(TvjX(t?) 2 G;X(t) 2 B)g?1
= ￡1?E(e?sTvjX(t?) 2 G;X(t) 2 B)?
￡fsE(TvjX(t?) 2 G;X(t) 2 B)g?1
x64x2excax2ex64x43x86x1bx8dx98x35x8cx1axb5
P(Tv? sjX(t) 2 B) =
Z s
0
P(Tv > ujX(t?) 2 G;X(t) 2 B)du
￡fsE(TvjX(t?) 2 G;X(t) 2 B)g?1
6.21
ePij = P(X(?n+1) = jjX(?n = i)
= P(X(?1) = jjX(0) = i)
=
‰ qij
qi j 6= i
0 j = i
eP =
0
@
0 3=5 2=5
1=4 0 3=4
1=3 2=3 0
1
A
P(N(t) = 1) = P(?1? t;?2 > t)
=
2X
i;j=0
Z t
0
P(?2?u > t?ujX(?1) = j;?1 = u;X(0) = i)
dP(?1? u;X(?1) = jjX(0) = i)P(X(0) = i)
= 23
Z t
0
e?5(t?u)d
1
4(1?e
4u)
+ 23
Z t
0
e?6(t?u)d
3
4(1?e
4u)
17
+13
Z t
0
e?5(t?u)d
1
3(1?e
6u)
+ 13
Z t
0
e?4(t?u)d
2
3(1?e
6u)
= 23e?5t(et?1)+e?6t(e2t?1)+ 23e?5t(1?e?t)+ 23e?4t(1?e?2t)
= 53e?4t? 43e?6t? 13
(2) x64xbdx6e6.10.3x1a
Φ1(s) = (s+q1)?1q10 +(s+q1)?1q12Φ2(s)
Φ2(s) = (s+q2)?1q20 +(s+q2)?1q21Φ1(s)
x64x64x8cx1a
Φ1(s) = s+12s2 +10s+12
Φ2(s) = s+8s2 +10s+12
xcfx8fP(T1 < t) = 23P(T1 < tjX(0) = 1)+ 13P(T1 < tjX(0) = 2)xa7xa4xb1P(T1 < t)x1bLaplacex43x86
x8fxb5
Φ(s) = 23Φ1(s)+ 13Φ2(s) = s+32=3s2 +10s+12
x64x64x8cx1axb5
E(T1) =?Φ0(0) = 71108
E(T21) = Φ00(0) = 307324
E(T31) =?Φ000(0) = 661324
E(T41) = Φ0000(0) = 5689972
(3) x17
= infft,t > 0;X(t) 6= X(0)g
E(T2) = E(T2?T1)+E(T1) = E(?jX(0) = 0)+E(T1)
= 1q
0
+ 71108 = 463540
P(X(T2) = 2) = P(X(?) = 2jX(0) = 0) = q02q
0
= 25
18
