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Chng 1. XC SUT CA BIN C
3.
a.
=. 1 + 2. (1 + 2 + 3) =. 1 + 2. 1 + 2 + 3 =+ 1 + 2 + 1 + 2 + 3 =+ 1 .2 + 1 .2 .3 b. =. 1 + 2. (12 + 23 + 13)7.
Gi l thi gian tu 1 n bn, 0;24Gi l thi gian tu 2 n bn, [0;24]
= ,:, 0;24 = 0;24x0;24A l bc hai tu phi i nhau
= , : x < < + 1 < < + 21 = , : x < < + 12 = , : y < < + 22 = , : x 2 < < = () = 242
12
232 12
222
242
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2
9
Gi th sinh vng , = 1,2,3H l bc th sinh c 3 vng
a.
= 123 = 1.(2/1)(3/12) = 0,9.0,8.0,7b.
2 = 1.(2/1) + 1 .(2/1 ) = 0,9.0,8 + 0,1.0(2 /) = 2 .
= 1 () =123 =1 +2 +3 2 1 +2 +3 2 . 1 +2 +3 =2 2 . = 2 1 +2 +3 = 2 = 1 (2)
24
24
1
2
y = x - 2
y = xy = x + 1
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3
10
Gi th sinh tr li ng cu trong 12 cu = 0..12
=
12(1
5)
. (
4
5)12
Gi l sim = 4 12 = 5 12a.
= 13, = 5
5
=
125 (
1
5)5. (
4
5)7
b.
< 0, < 125
= 0,1,20 + 1+ 2
11
Do xs bn 3 vin trng vng 10 l 0,008 nn xs 1 vin trng vng 10 l 0,2
( V gi l xs 1 vin trng vng 10 th xs 3 vin trng vng 10 l 0,008 nn 3 = 0,008 suy ra = 0,2)Xs 1 vin trng vng 9 l: 1-0,2-0,15-0,4=0,25
Gi l bc c vin trng vng 10 v 3 vin trng vng 8 = 3(0,2) .(0,15)3 Gi
l bc c
vin trng vng 10 v 3
vin trng vng 9
= 3(0,2) .(0,25)3 A l bc c t nht 28 im
= 3 + 2+ 2 + (1)12
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4
Gi l bc c ng c ca cnh phi h, = 0..3 = 3(0,9) . (0,1)3 Gi
l bc c
ng c ca cnh tri h,
= 0. .2
= 2(0,95) . (0,05)2 a.
Gi l bc my bay hot ng = 1 = 1 10 01 0(0)
b.
= 1
= 1
0+
0= 1
0 0+
0.
(
0)
13.
Gi l bc chn c chi tit do my th sn xut, = 1,2.H l bc chn c chi tit t tiu chun
a.
= 1.(/1) + 2.(/2) = 13 0,65 + 23 .0,8
b.
2/ = 2. = (2)(/2)() = (2)1 (/2)1 () 14
Gi
l bc c
sp loi A trong 2 sp ly ra,
= 0,1,2
l bc sp ly sau cng l sp loi A = 0.(/0) + 1.(/1) + 2.(/2)0 = 212 . 420 ,1 = 1012 . 420 + 212 . 1620
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5
2 = 1012 . 1620
(
/
0) = 0,
(
/
1) =
1
2,
(
/
2) = 1
15
Gi l bc ly c bt t hp 1, = 0,1Gi l bc ly c bt t hp 2, = 0,1,2Gi l bc ly c bt b sang hp 3, = 0,1,2,3a.
A l bc trong hp 3 c sbt nhiu hn s bt xanh
= 3 = 12 = 1.2 = 515 . 32102 b.
Gi l bc ly c 2 bt cng mu t hp 3. = 0.(/0) + 1.(/1) + 2.(/2) + 3.(/3)
3 =5
15 . 32
102 ,2 = 1.1 + 0.2 = 5153171102 + 1015 72102 1 = 1.0 + 0.1 = 515 72102 + 10153171102
0
=
0
.
0
=
10
15
72
102
(/0) = 32 + 72102 (/1) = 42 + 62102
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(/2) = 52 + 52102
(
/
3) =
62 +
42
102
17
Gi l bc chn c x th nhm , = 1,2,3,4 l bc chn c x th bn trng.a.
=
1.
(
/
1) +
2.
(
/
2) +
3.
(
/
3) +
4.
(
/
4)
1 = 625 ,2 = 725 ,3 = 825 ,4 = 425(/1) = 0,8,(/2) = 0,7,(/3) = 0,6,(/4) = 0,5b.
So snh: (1/),(2/),(3/),(4/)18
a.
Gi thu c ln trong 4 ln pht = 4(0,4) . (0,6)4 A l bc thu tn hiu
= 0 = 40(0,4)0. (0,6)4 = (0,6)4
= 1
(
)
b.
Gi thu c ln trong ln pht = (0,4) . (0,6) A l bc thu tn hiu
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0,95 1 0,95
0,05
= 0 = 0(0,4)0. (0,6) = (0,6) (0,6) 0,05 log0,6 0,05 = 5,86 = 6
19, 20. Tng tnh bi 18.
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Chng 2 I LNG NGU NHIN
1.
a.
Theo tnh cht ca bng phn phi xc sut th phi tha mn: , 2, 3, 2, 22, 72 + [0,1] + 2 + 2 + 3 + 2 + 22 + 72 + = 1Tng ng
, 2, 3, 2, 22, 72 + [0,1]102 + 9 1 = 0
, 2
, 3
,
2 , 2
2, 7
2 +
[0,1]
= 1 = 110
Vy = 110
b. 5 = = 5 + = 6+ = 7c. = 32. 0,1,2
= 0 = 82102 , = 1 = 2181102 , = 2 = 22102 X 0 1 2
P 82
102
2181
102
22
102
=
0 082102 0 < 182 + 2181102 1 < 21 > 2
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3.
a. 0,1,2,3
= 0 = 63
163 , = 1 = 101
62
163
= 2 = 102 61163 , = 3 = 103163 X 0 1 2 3
P 63163 101 62163 102 61163 103163 b.
=. 5 + 3 . 8 = 24 3
( = 24) = 0 = 63163 = 21 = = 1 = 101 62163 = 18 = = 2 = 102 61163 = 15 = = 3 = 103163
Y 24 21 18 15
P 63163 101 62163 102 61163 103163 4.
Gi
l bc c
bi t hp 1 chuyn sang hp 2,
= 0,1.
Gi l bc c bi t hp 2 chuyn sang hp 1, = 0,1,2. l sbi hp 1 sau 2 ln chuyn. = 0 = 10 = 1.(0/1) = 113142 .3262
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10
= 1 = 11+ (00)11 = 1(1/1) = 1131
42 .
3131
62
00 = 0(0/0) = 3242 4262 = 2 = 12 + (01)12 = 1(2/1) = 113142 .326201 = 0(1/0) = 32
42
2141
62
= 3 = 0202 = 0(2/0) = 3242 2262X 0 1 2 3
P 113142 .3262 113142 .313162+
32
42 42
62113142 .3262+
32
42 21
41
62 3242 2262
l sbi hp 2. = 3 Y 3 2 1 0
P 1131
42 .
32
62
1131
42 .
3131
62
+ 32
42 42
621131
42 .
32
62
+ 32
42 21
41
62 32
42
22
62
5.
2 0 1 2
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P 0,3 0,5 0,2
X + Y -2 -1 0 1 2 3
P 0,2.0,3 0,2.0,4+0,3.0,3 0,3.0,4 0,3.0,4+0,3.0,3+0,2.0,3 0,3.0,3+0,2.0,4 0,2.0,3
2Y -2 0 2
P 0,3 0,4 0,3
X Y -2 -1 0 1 2 3
P 0,2.0,3 0,2.0,4+0,3.0,3 0,2.0,3+0,3.0,4+0,3.0,3 0,3.0,3+0,3.0,4+0,2.0,3 0,2.0.4 0,2.0,3
XY -2 -1 0 1 2
P 0,2.0,3 0,2.0,3+0,3.0,3 1 0,7.0,6 0,2.0,3+0,3.0,3 0,2.0,3
6.
Max(X1,X2) 1 2 3 4 5 6
P 1/36 3/36 5/36 7/36 9/36 11/36
7.
Gi l bc ln th th mc ca. l s ln th cn thit = 1 = 1 = 25
= 2 = 12 = 1 (2/1 ) = 35 . 24 = 3 = 1 .2 .3 = 1 (2 /1 )(3/1 .2 ) = 35 . 24 . 23 = 4 = 1 .2 .3 .4 = 1 (2 /1 )(3 /1 .2 )4/1 .2 .3 = 35 . 24 . 13 . 1
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Bng phn phi xc sut
X 1 2 3 4
P 2
5
3
5.2
4
3
5.2
4.2
3
3
5.2
4.1
3. 1
Suy ra EX, DX
8.
Gi l s ln dnga.~(3; 2/5)
= 3.
2
5
=6
5
,
= 3.
2
5
.3
5
=18
25
b. Gi l bc dng n th = 0 = 1 .2 .3 = 1 .2 .3 = 12 . 38 . 23 = 1 = 1.2 .3 + 1 .2.3 + 1 .2 .3
= 1 = 12
.3
8.2
3+
1
2.5
8.2
3
= 2 = 1.2.3 + 1.2.3 + 1 .2.3 = 3 = 1.2 .3 =Ta c bng phn phi xc sut v t tnh c .9.
Cch 1: Do,c lp nn + = + , + = +
=
.
,
=
2
2
(
)2 =
2
2
(
)2(
)2
Cch 2: Tng tnh bi 5, ta lp bng phn phi xc sut cho + v ri tnh.10.
Tng tnh cc v d dy trn lp.
11.
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a.
+
= 1 = 12
b.
= =
0 = 0 < /2 02
+ 12 .
2
=1
2 + 1
2;2
1
>
/2
c.
= 0
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0
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Chng 3. CC QUY LUT PHN PHI XC SUT THNG DNG
1.
~
(700;250; 80)
Nn theo tnh cht ca quy lut siu bi ta c:
= = 80. 250700
= 1 = 80. 250700 . 450700 . 700 80700 1 2.
a.
3
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4.
Gi l s ht thc ging b lp trong 1000 ht th~(1000; 0,006). Do = 1000 0, =0,006
0 nn
xp x bi Poisson vi
=
= 1000.0,006 = 6.
= = 6 6! , a.
( = 6)b.
3 = 1 ( < 3)c. ( 5)5.
Gi l s b trai c trong 200 b th~(200; 0,51). Do = 200 0, = 0,51 khng qugn 0 v gn 1 nn xp x bi quy lut chun vi = = 200.0,51 = 102,2 = =200.0,51.0,49 = 49,98.
(102; 49,98)
< 200 = < 100 = 100 10249,98 = 2,8 = 1 2,86.
Gi l s cu tr li ng,~(45; 14). = 45 0, = 1
4khng qu gn 0 v gn 1 nn
xp x bi quy lut chun vi = = 45.1/4 = 11,25,2 = = 45.1/4.3/4 = 8,4375.
(11,25; 8,4375)
a.
PX = k 1npqk npnpq , vi x = 12 ex22 hm Gauss( = 30)
b.
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23 = 1 23 11,258,4375 = 1 4 = 0c.
20 = 20 11,258,4375 = 3.1 = 17.
Gi l s bng loi trong my tnh b hng = 1,2,3.1~(1000;0,0005) nn xp x1 1000.0,0005;1000.0,0005.0,9995 = (0,5; 0,49975)2~(3000;0,0003) nn xp x2 3000.0,0003;3000.0,0003.0,9997 = (0,9; 0,89973)3~(6000;0,0001) nn xp x3 6000.0,0001;6000.0,0001.0,9999 = (0,6; 0,59994)
Nn1 +2 +3 0,5 + 0,9 + 0,6; 0,49975 + 0,89973 + 0,59994 = (2; 1,99942) l bin c my ngng lm vic
= 1
= 1
1 +
2 +
3 < 2
= 1
2
2
1,99942= 1
0
= 0,5
8.
= 0,2
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= = < 0,588 = 0,588
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0,9(1 )
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Chng 4. I LNG NGU NHIN NHIU CHIU
1.
a.
XY
0 1
0 2
51
4
3
52
4
8
20
1 2
53
4
3
52
4
12
20
8
20
12
20
b.
X 0 1P 8/20 12/20
Y 0 1
P 8/20 12/20
2.
I. Tng tbi 5 chng 2
II. Z = 2X Y +5
YX
-1 0 1
1 Z=80,1
Z=7
0,15Z=6
0,052 Z=10
0,3Z=9
0,2Z=8
0,2
Vy
Z 6 7 8 9 10
P 0,05 0,15 0,2+0,1=0,3 0,2 0,3
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3. Tng t 2
4.
a. chng minh
,
c lp ta kim tra iu kin
= ; = = = . = , 1,2, 1,2,3 = ; = Da vo bng phn phi ng thi ca (,). = da vo phn phi l ca. = da vo phn phi l ca .b. Tng t v d cha.
5.a. Do,c lp nn
= ; = = = . = , 0,1,2,3, 0,1,2,3,4Suy ra bng phn phi ng thi ca (,).b.Da vo bng phn phi ng thi ta c: > = = 1, = 0+ = 2, = 0 + = 3, = 0+ = 2, = 1
+ = 3, = 1+ = 3, = 26. Tng t v d cha trong slides chng 4
7.a. Tng t v d cha trong slides chng 4
b. Tng t bi 4.a
8.b. 1,2,3. Da vo cc bi trong hp 1 ta c bng phn phi caX 1 2 3
P 1/6 2/6 3/6 1,2,3. Da vo cc bi trong hp 1 ta c bng phn phi ca X 1 2 3P 2/6 2/6 1/6
c. Suy ra
,
a. Do,: = ; = = = . = , 1,2,3, 1,2,3Suy ra bng phn phi ng thi ca (,).9, 10, 11, 12 Tng tnh v d cha trong slides chng 4.
13. Tng t bi 4
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14. Tng tnh v d cha trong slides chng 4.15.a. Da vo tnh cht:
0
1
=1=1 = 1Ta tm c mb. Tng tnh v d cha trong slides chng 4.
16.a. Tng t bi 1b.-Tm bng phn phi c iu kin ca
theo
= (
= 0) ri tnh kz vng theo bng ny.
-Tm bng phn phi c iu kin ca theo = ( = 2) ri tnh kz vng theo bng ny.c. Tng tnh v d cha trong slides chng 4.d. Z= Y/XX
Y100 150 200
0 Z=00,1
Z=0
0,05Z=0
0,051 Z=1/100
0,05Z=1/150
0,2Z=1/200
0,15
2 Z=2/1000
Z=2/150
0,1Z=2/200
0,3
Vy
Z 0 1/200 1/150 1/100 2/150 2/100P 0,2 0,15 0,2 0,35 0,1 0
17, 18,19. Tng t bi 16
20.1. Da vo bng phn phi l ca ta c: = 0. 0,45 ++ 1. 0,25 + = 0,5Nn = 0,25Mt khc do =1=1 = 1 nn
0,2 + 0,25 + + + 0,15 + 0,1 = 1Nn = 0,05Suy ra
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X 0 1P 0,5 0,52. Tng t cc bi trn.
21.a. l chi ph sa cha ca mt ng c(/v triu ng/nm)Ta c bng:X 0 5,5 7,2 12,5
P 0,9 0,02 0,05 0,03Chi ph sa cha trung bnh ca mt ng c: EX = 0,845 (triu ng/nm) l chi ph sa cha ca mt ng c (/v triu ng/nm)Ta c bng:Y 0 5,5 7,2 12,5P 0,9 0,01 0,04 0,05
Chi ph sa cha trung bnh ca mt ng c : EY = 4,055 (triu ng/nm)Do gi bn hai loi ng c trn l nh nhau nn ta s chn ng c c chi ph sa cha thptc l chn ng c A.b.Gi l chi ph sa cha ca cng ti th = 6 + 4. Do,c lp nn chi ph sa chatrung bnh ca ca cng ti l = 6 + 4.22.1. ,c lp ta kim tra iu kin
= ; = = = . = , 0,1, ,0,1xc nh , .2. Da vo gi thit , = 0ta tm c . Sau , da vo bng phn phi ton phnca (,) kim tra tnh c lp.23.1. Xc nh bng phn phi xc sut ca vi iu kin = = , = 1,2.- Nu thy ( =/ = ) = ( =)ng vi mi , = 1,2 th,c lp.2.
. 3 = 1 . > 3 = 1 . = 4 = 1 = 2, = 2 = 1 1
2 =
1
2 + > 2 = 1 + 2 = 1 = 1, = 1 = 1 18 = 78
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24.1. Bng phn phi ng thi ca (
,
)
X
Y
1 2 3 4 5
1 1/15 1/15 1/15 1/15 1/152 0 1/15 1/15 1/15 1/15
3 0 0 1/15 1/15 1/15
4 0 0 0 1/15 1/155 0 0 0 0 1/15
Da vo bng ta thy,khng c lp2. Tng tnh bi 23.1
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Chng 5. BI TON C LNG
1.
Chn nn 1 sp v gi
l s sp loi 1 chn c
~() = 1 = l t l sp loi 1 ca nh myc lng t l sp loi 1 ti thiu ca nh my th s dng cn di ca min tin cy bnphi, tc l
= (1 )
= 500, = 360, = 0,72, = 0,05, = 1,652. Tng tbi 1 nhng s dng khong tin cy i xng.3. Tng t VD trong ti liu (c lng s c c trong h)
4. cha
Gi N l s c c trong h. Bt nn 1 con c trong h v gi X l sc nh du bt c. X cquy lut A(p)
PX = 1 = p = 2000N l t l c bnh duff1 fn u2 < = 2000N
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Doanh s ti thiu bng . 0,8.220002000
= = + (1 )
= 0,233
= 20000,233 = 8 583,69 8 584Doanh s ti thiu bng
8 584.0,8.22000 = 151 078 400
5.
2 = 2.1 . /2 = 0,02 = 0,9, = 0,01;0,005 = 2,571
0,01. /22 = = 5,9444 = 6
6. Hng dn: c lng = + 7. Hng dn:c lng ; + , = /28. 1 tng tbi 7 (nhng cha bit, tc l = /2) 2 tng t bi 5.
9. Tng t bi 5.
10.
a. Tng t bi 7
b. = /2
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c. Tng t bi 5.
12.
a. Tng t bi 7
13.
a. Tng t bi 7
14.
a. = c c lng khng chch l b. 2 = c c lng khng chch l 2 c. l t l tri c khi lng khng qu 95 gam c c lng khng chch l t l mu = 15a. c lng trung bnh ging bi 7 (Da vo ton b bng)
b. c lng t lt chun ging bi 2
c. c lng trung bnh ging bi 7 (Da vo bng ch gm cc chi tit t chun)
d. Tnh
=
/2
2
e. Tng t bi 5.
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Chng 6. BI TON KIM NH GI THUYT THNG K1.
Chn nn 1 sp v gi l s phphm chn c.~() = 1 = l t l phphm 0: = 0,05; 1: < 0,05 = (; ) = (;0,05 ) = (;1,65)
= 0,050,05.(1 0,05) = 100, = 8, = 0,08
= 1,376 Kt lun: Tha nhn 0 , bc b1 tc l l hng khng t tiu chun.2. Tng t 1 vi 0: = 0,06; 1: > 0,063. Tng t 1 vi 0: = 0,8; 1: > 0,84. Tng t 1 vi
0: = 0,2; 1: < 0,2 = 40.10 = 400, = 1.2 + 2.0 + 3.4 +5.
Chn nn 1 cng nhn x nghip A v gi l s cng nhn b tai nn chn c.~(1)
= 1
=
1l t l cng nhn b tai nn A
Chn nn 1 cng nhn x nghip B v gi l s cng nhn b tai nn chn c.~(2) = 1 = 2 l t l cng nhn b tai nn B0:1 = 2;1: 1 2
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= ;2
2; + , = 0,02
= ;2,33 2,33;+ = 1 2(1 )11 + 121 = 400,1 = 20,1 = 0,052 = 500,2 = 28,2 = 0,056= 1 +21 + 2 = 0,053
=
6. Tng t bi 5 vi
0 :1 = 2;1: 1 > 27.
Chn nn 1 sp v gi l trng lng ca sp ny
=
l trng lng TB ca sp.
0: = 500;1: < 500 = ;1 = ;0,0324 = (;2,064) = 500
8. Tng t bi 7 vi
0:
= 86,5;
1:
86,5
9. Tng t bi 7 vi 0: = 150;1: > 15010. Tng t bi 9 vi
0: = 4,5;1: > 4,5
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(Ch vn vo)
11.
Chn nn 1 gia nh trong khu vc v gi
l nhu cu v loi hng ca gia nh ny trong nm.
= l nhu cu trung bnh ca mt gia nh v loi hng ny trong nm.4000. l nhu cu trung bnh ca khu vc v loi hng ny trong nm.
0:4000 = 168;1: 4000 1680: = 1684000 ;1: 1684000
Tng t bi 7.
12.
a.
Chn nn 1 sp ca phn xng 1 v gi1 l trng lng ca sp ny1 = 1 l trng lng trung bnh ca sp ca phn xng 1Chn nn 1 sp ca phn xng 2 v gi2 l trng lng ca sp ny
2 =
2 l trng lng trung bnh ca sp ca phn xng 2
0: 1 = 2;1: 1 2 = ;2
1+22 2
1+22; + = ;0,025900+8002 0,025900+8002; +
= ;1,96 1,96;+
=
1
2
1 112 + 2 1221 + 2 2 11 + 12
b.
Chn nn 1 sp ca phn xng 1 v gi 1 l s phphm chn c
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1 = 1 l t l phphm ca sp ca phn xng 1Chn nn 1 sp ca phn xng 2 v gi 2 l s phphm chn c
2 =
2 l t l phphm ca sp ca phn xng 2
0:1 = 2;1: 1 2 = ;/2 /2; + = ;0,005 0,005 ; + = ;2,57 2,57;+
Gqs =f1 f2
f
(1
f
)
1
n1+
1
n2
1 = 900,1 = 18,1 = 0,022 = 800,2 = 16,2 = 0,02 = 0 Tha nhn 0, bc b1 tc l c th coi t l phphm ca 2 phn xng ny l nh nhau.