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#%$%&%(%)%*%+-,-.0/
2、
{1{
#%1%&%(%2%3%+-,04-5
61.
798:<;=>p?A@BC;GFDDEEH@I
pk = P ( = k) = qk 1p; k = 1; 2; :::
FKJq= 1 p.
2.6
7(1), L = 2 M;ONP( = 2) = pq + qp = 2pq, PRQSfUWVTXYOZ[UWXgAG\]VY^a` L = 3 M;GNP( = 2) = p2q + q2p PRQSfUWVTVXYGZbUWXgA\
3、7]XV^Y
cd;Le = k M;
P ( = k) = pk 1q + qk 1p = pq(pk 2 + qk 2; k = 2; 3; 4; :::
(2), L = k M;GfKgihkjkl(FV\m]p)@;Gnk
1 jJiNr 1 jklVm
;Gr
PFopm ^
11pr 1(1 p)k
1
(r
1) = Ckr
11pr qk r ;
pk = P ( = k) = pCkr
k = r + 1; r + 2; :::;
FK
4、Jq= 1 p.
3.6
7(1), L 1 = k M;GfKgiskA5tu?vwkx;GFA4@ouy k +z1; :::; 10, r
C 4
P ( 1 = k) =
10 k
;
k = 1; 2; :::; 6:
C 5
10
L 3 = k M;OfKgiskA5tu?Nvu{y 1; z2;:::; k
1 ;OF2ouyk +z1;
5、:::; 10, r
C 2
1
C 2
k
P ( 3 = k) =
k
10
; k = 3; 4; :::; 8:
C 5
10
5
(2), |M}~10`Gu????@f1=S1gT= f5 jt?KJic??j?N1g ;Gr
1
5
P ( 1 = 1) =
C5i
95 i :
X
105
i=1
?STf 1 = 10g = f5 jt????10gt;G
6、|?
1
P ( 1 = 10) = 105 :
{1{
L k = 2; :::; 9 M;??i?
?
f 1 = kg = f 1 kg f 1 k 1g;
1
5
P ( 1k)
=
X
C5iki(10
k)
7、5
i ;
10
i=1
8、
1
5
P ( 1k
1)
=
X
C5i(k
1)i [10
(k
1)])5 i :
10
9、
i=1
?q?
P ( 1 = k) =
1
5
k)5
i
(k
1)i (11
k)5
i]:
C5i[ki (10
X
10、
10 i=1
4.6
7 2 f2; 3; :::; 12g, FDE?@
11、
2
3
4
5
6
7
8
9
10
11
12
P ( = k)
1
2
3
4
5
6
5
4
3
2
1
12、
36
36
36
36
36
36
36
36
36
36
36
5.6
7??Bn(m)
@N
j?N@j??M????;G?
m j??M??n;Am
m
13、
n
P ( = n) =
X
P (Bn(m)jAm )P (Am )
m=1
14、
n
11pmqn m 1(1
1
=
m=1 Cnm
w )m
X
15、
n
11pk+1qn k 1
1
=
k=0 Cnm
1 (1
w )k+1
X
1
16、
p
=
p
p(1
)[1
)]n 1
w
w
17、
p
p
p
= p 1 (1
)n 1 + (1
)n 1
(1
)n ;
w
w
w
FKJk= m
1; p = 1
p.
18、
67??@K?iA???;G?2jf1;?2; :::g. ? B @K?i??kj;Gnk
1 j??Ji;Ghkj
.
? ?Ji;C ?i??kj;ink
1 jK;???Ji;ihkjK?i??JJ`iAk@Kh?k
j??Ji;MBk @ihk j??J`?M
19、
1Ak
f = kg = B + C = A1B1 A2B2 :::Ak 1Bk
+ A1 B1A2 B2:::Ak
1 Bk
1Ak Bk :
{2{
?i
pk = P ( = k) = P (B) + P (C )
=
(0:6 0:4)k 10:4 + (0:6 0:4)k 10:6 0:6
=
(0:24)k 10:76 = qk 1p; k = 1; 2; :::
Q <=>p?0:76@ ABC7D
20、E
? @A??;Gj?2?f0; 1; 2; :::g. STf = 0g @Kh?cj?Ji;Gr
p0 = P ( = 0) = 0:4:
k 1 M;
;
f = kg = A1 B1A2 B2
:::Ak
1 Bk
1Ak Bk + A1 B1 A2B2
:::Ak Bk Ak+1
?qN
pk = P ( = k) = (0:6 0:4)k 1 0:6 0:6 + (0:6 0:4)k
21、 0:4
=
(0:24)k 10:456;
k = 1; 2; :::
??; ADqEBCDE`
7.
7(1) g (x) =
1
x+h F (t)dt ?I
1
0
0
h
Rx
) = 0, 3
0
; (x
0)(x)().
, ;2; (+1) = 1; (
10 , L x >
22、y M;= x
y (> 0), ?
Z
F (t)dt#
(x)(y) =
h "Zx
F (t)dt
y
1
x+h
y+h
Zy
F (t)dt# :
=
h "Zy+
F (t)dt
1
y+ +h
23、
y+h
< h, ?
"Zx
Zy
F (t)dt#
(x)(y) =
h
F (t)dt
1
x+h
y+h
=
"Zy+
F (t)dt
Zy
F (t)dt
Zy+
F (t)dt#
24、
h
1
y+ +h
y+
y+h
=
"Zy+h
F (t)dt
Zy
F (t)dt#
h
1
y+h+
y+
= F (y + h + 1 ) F
25、 (y + 2 ) 0; ( ?iy?+ h + 1 > y + 2 )
FKJ0< 1; 2 < 1, wcuJi`D
{3{
0 < h ;G?(x)?(y) = F (y + + 1h) F (y + 2h) 0 `
L h < 0 MgiQ8h;; (x) 7
20 , ?DJi(x)= F (x + h), FKJ0< < 1 ;GK?
8
lim
F (x) = 1
;
x!+1
<
x lim
F (x) = 0
:
!
Q
26、
8
lim
(x) = 1
<
x!+1
(x) = 0
:
3 ,
(x) = F (x + h),
F :
lim
x!
0
?
?i?(x) ;G?
lim
(y) = lim F (y + h) = F (x + h) = (x);
y!x
y!x
Q`
8.
7(1) ?i?
N
Z +1 1 dF (y) = 1 F (y)j+1 + Z +1 1 F
27、 (y)dy =
xyyxxy2
x!+1 x Z
+1 1
x!+1
F (x)
xy dF (y) =
x
lim
lim
x
=
lim
[
F (x)] +
x!+1
x F (x) + Z
+
y2 F (y)dy;
x 1
1
1
+
1 y2 F (y)dy
+ Zx
1
28、
R
+1
1
F (y)dy
x +
2
x
1
lim
y
! 1
x
1
F (x)
2
=1 + lim
x
= 1 + 1 = 0:
1
x!+1
x2
(2) gi(1) ?;
29、Gr`
(3) L x > 0 M; Rx+1 y1 dF (y) < +1, ??7Rx+1 y1 dF (y) = +1, ?K?i?
Z
+
1
y dF (y) <
Z
+
1
y2 F (y)dy;
x
x
1
1
R
+
1
1
F (y)dy = +1, ?qN
x
y2
+
1
+
30、1
1
x 1
F (y)dy
lim
x
dF (y) =
lim [
F (x)] +
lim
y2
R
1
x
!
0+
Z
x
y
x
!
0+
x
!
0+
x
1
31、
F (x)
2
=
lim [
F (x)] +
lim
x
1
x!0
+
x!0
+
32、
x2
=
lim [
F (x)] +
lim
F (x) = 0:
x!0+
x!0+
{4{
610.
7??( ; ) q A Jic;G?( ; ) A p.d.f.
@
f (x; y) = (
1
; (x; y)
A
= (
1
; 0 < y < 2x
33、 x2; 0
x
2
0;
F2
0;
F
(A)
(A)
FKJ(A) = R
02(2x x2 )dx = (x2
1
x3 )j02 =
4
. ?|? A?p.?d.f.
3
3
Z
(
0;
F
1 f (x; y)dy =
3
(2x
34、 x2); 0x2
f
(x) =
4
6–
IGL0 x2 M; ADE—?@
3
x
1
Z0
F (x) = P ( < x) =
(2t t2)dt =
(3x2x3);
35、
4
4
Q
F (x) =
f (x) =
8 4
(3x2
x3); 0 < x2 ;
>
0;
x
0
1
<
1;
x > 2
>
36、
(2x
x ); 0
x
2
:
=
Z
1
f (x; y)dy =
4
:
dx
(
0;
2
F
dF (x)
3
611.
7??ABC A“h@;G”@‘AB `
37、?L 0 < x < h M; ADE—?@
1
1
2
F (x) = P ( < x) =
2
ABh
2
AB(h
x)
;
1
2
ABh
’? A @;?B@VcK?i?;Gfifly= x ABC A?0;B@0 ;G|?
A0B==AB =
h x
;
A0B0
= AB
h
x
;
h
38、
h
(h
x)2
F (x) = 1
:
h2
Q
613.
8
> 0;
<
F (x) = 1
>
: 1;
(h x)2
x0
;
0 < xh :
h2
x > h
75,P ( k) = P (1
39、k) = P ( 1 k) = 1 P ( < 1 k) = 0:25, ?| P ( < 1 k) =
0: r 1 k = 0; 29; k = 0:71.
{5{
15.
7? P( ). ?K?\i?]?
1
X
pk = P ( = k) = P ( = kj = j)P ( = j):
j=k
?i‰? j = j B(j; p), r P ( = kj = j) = Cjk pk (1
p)j k ;G
1
k
k
j k j
pk
40、 =
X
Cj p (1
p)
r
j=k
j!
1
j!
k
j k j
=
X
p (1 p)
r
j=k
k!(j
k)!
j!
41、
=
k pk e
1
j k(1 p)j k
X
k!
j=k
(j
k)!
k pk
=
e
k!
( p)k
=
e
k!
Q P( p)
19.6
7??F(x) Alim F (x) = A = 1,
x!1
e (1 p)
p; k = 0;
42、 1; 2; :::
Q A = 1,
f (x) = F 0(x) =
( 0;F
2x; 0 < x < 1
620.
7(1),
Z x
F (x) = P ( < x) =
f (t)dt
8 2 x2
8 2 x2
;
0 < x1
;
0 < x1
>
0;
x
0
>
0;
x
0
1
43、
1
=
1
1
2
3
=
1
2
>
+ 2x
x
; 1 < x2
>
2x
x
1; 1 < x2
2
2
2
2
>
>
<
1;
x > 2
<
1;
x
44、 > 2
>
>
>
>
>
>
:
:
(2) P ( < 0:5) = F (0:5) =
1
0:52
= 0:125.
45、
2
P ( > 1:3) = 1 F (1:3) = 1
2 1:3 +
1
(1:3)2 + 1 = 0:245;
2
P (0:2 < < 1:2) = F (1:2) F (0:2) = 2 1:2
1
(1:2)21
1
(0:2)2 = 0:66:
2
2
{6{
21.6
46、
7(1)
1
1
1
F1 (y) = P ( 1 < y) = P
< y = P
>
= 1 F
y
y
dF1 (y)
1
1
47、
0
1
1
f1(y) =
= f
=
f
dy
y
y
y2
y
1
2
; 0 <
1
< 1
= (
2
; 0 < y <
1
=
3
y2
(
48、0;
F
0;
F
y
y
y
(2)
(
(
2
j
j
P ( y < < y); y > 0
F (y) F ( y); y > 0
F
(y) = P (
< y) =
0;
49、
y0
=
0;
y0
2
dy
(
f
(y) + f ( y); y > 0
( f (y); y > 0
f
(y) =
dF1 (y)
=
0;
y0
=
0;
y0
=
8 2y; 0 < y < 1 = ( 0;F
50、
>
0;y0
2y; 0 < y < 1
<
0;
y1
>
(3)
:
51、
F3(y) = P e < y = P (< ln y)
= P ( > ln y) = 1 F ( ln y):
f3(y) =
dF1 (y)
=
1
f (
52、 ln y)
dy
y
(
2 ln y
;
0F
= (
2 ln y
=
0; y
0; y
< ln y < 1
22.6
7(1) A??DE?@
; e 1 < y < 1
F
n
X
pn = P ( = n) = pnm
m=0
ne
n
n
=
Cnmpm (1 p)n m =
e ;
53、n = 0; 1; 2; :::
X
n!
m=0
n!
Q P( ).
(2) A??DE?@
1
X
pm = P ( = n) = pnm
n=m
{7{
Q P( p).
23.6
7(1)
f (x) =
f (y) =
pm me
1
[ (1
p)]n m
=
X
m!
n=m
(n
m)!
pm
54、 me
e (1
p) =
( p)m
=
e p; m = 0; 1; 2; :::
m!
m!
+
f (x; y)dy =
(
+1
x
1
= (
x
0;
xe
(1+y)
2 dy;
x
0
0;
;
x
0
Z
1
0
x > 0
xe
x >
0;
+
(
R
+1
x
55、
1
= (
1
f (x; y)dx =
xe
(1+y)
2 dx;
y
0
0;
2
y
0
0;
Z
1
R
0
y > 0
(1+y) ;
y >
0;
?@ f (x; y) = f (x)f (y), 7
56、
(2)
f (x)
=
+1
f (x; y)dy =
x1 8xydy;
0x < 1
=
(
4(x
x3 ); 0x < 1
Z
(
0;
F
0;
F
R
f
57、(y)
=
+1
f (x; y)dx =
0y 8xydx;
0y < 1
=
(
4y3; 0y < 1
Z
(
0;
F
0;
F
R
?@ f (x; y) 6= f (x)f (y), 7
(3)
f (x; y)dy = (
58、
f (x) =
0;
k1 ) k2 )
x0
Z
+
1
R
+1
1
x
k1
1
(y
x)
k2
1
e
y
dy;
x > 0
x
+1
1
1
59、
(
xk1
1tk2
1e x e
tdt;
x > 0
= (
xk1
1e x; x > 0
=
k1 )
k2 )
0;
k1 )
0;
x
0
x
0
R
0
60、
Q
624.
7(1)
f (x)
f (y)
k1 ; 1), k2 ; 1), ?@ f (x; y) =6 f (x)f (y), 7
( 0;
F
( 0;
2x
F
=
+1
f (x; y)dy =
R
2
x2 +
xy
dy; 0x1
=
2x2 +
;
0x1
0
3
3
Z
( 0;
F
( 0;
F
=
+1
61、f (x; y)dx =
R
1
x2 +
xy
dx; 0y2
=
1
0y2
0
3
6 (2 + y);
Z
(2) L 0 y 2 M;
f (x; y)
f j (xjy) = f (y)
{8{
(3)
(4)
FKJ
8
x2+
xy
62、
(
6x2 +2xy
3
0;
F
0;
F
<
2+y
;
0x1
=
61
(2+y) ;
0x1
63、
=
:
P ( + > 1)
=
P (( ; ) 2 D) = Z
ZD f (x; y)dxdy
1
2
xy
64、
=
Z0
dx Z1 x x2 +
dy
3
1
x2(1 + x) +
x
[4 (1 x)2 ] dx
=
Z0
6
1
5
4
1
65、
65
=
Z0
x3 +
x2 +
x
dx =
6
3
2
72
1
1
=
P<
1
; <
1
;
P<
<
2
2
66、
2 j
2
P<
1
2
1
1
2
應(yīng)用概率統(tǒng)計(jì) 課后答案
