应用概率统计 课后答案
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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 jtKJic jN1g ;Gr 1 5 P ( 1 = 1) = C5i 95 i : X 105 i=1 STf 1 = 10g = f5 jt10gt;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 7Bn(m) @N j?N@jM;G m jMn;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@KiA;G2jf1;?2; :::g. B @Kikj;Gnk 1 jJi;Ghkj . Ji;C ikj;ink 1 jK;Ji;ihkjKiJJ`iAk@Khk jJi;MBk @ihk jJ`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;Gj2?f0; 1; 2; :::g. STf = 0g @KhcjJi;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, Ki 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 . | AŒp.œd.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. 7ABC 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 @;ŸB@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) AŒœDE@ ; 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) AŒœDE@ 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
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