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PBT501S - PROBABILITY THEORY1 - 2ND OPP - JULY 2023 |
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n Am I BI A u n IVER s I TY
OF SCIEnCE Ano TECHnOLOGY
FACULTY OF HEAL TH, NATURAL RESOURCES AND APPLIED
SCIENCES
SCHOOLOF NATURALAND APPLIEDSCIENCES
DEPARTMENTOF MATHEMATICS,STATISTICSAND ACTUARIALSCIENCE
QUALIFICATION: BACHELOROF SCIENCEAPPLIEDMATHEMATICSAND STATISTICS
QUALIFICATIONCODE: 07BSAM
COURSECODE:PBT501 S
LEVEL:5
COURSENAME: PROBABILITY THEORY I
SESSION:JULY 2023
PAPER:THEORY
DURATION: 3 HOURS
MARKS: 100
SUPPLEMENTARY/SECONDOPPORTUNITYEXAMINATION QUESTION PAPER
EXAMINER(S) Dr. D. Ntirampeba
Mr. E. Mwahi
MODERATOR: Mr. A. Roux
THIS QUESTIONPAPERCONSISTSOF 4 PAGES
(Excluding the cover page)
INSTRUCTIONS
1. Answer ALLthe questions.
2. Write clearly and neatly.
3. Number the answers clearly.
PERMISSIBLEMATERIALS
1. Non-programable calculator
ATTACHMENTS
1. Statistical tables (Z-Table)
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Question 1 [25 marks]
1.1
Assume that { Bi,B2 , ••• ,Bk}is a partition of S such that P(B;)> 0, for i = I, 2,..., k.
k
Provethatforanyevent A in the samples, P(A)=IP(AIB;)P(B;)
[SJ
i=l
1.2
Let X be a binomial random with a probability mass function given by
f(:) f(x) = pkqn-x,for x = 0,1, ..., n
l
0, elsewhere
Show that E(X) = np.
[10]
1.3
Dr.Richmond, a psychologist, is studying the daytime television viewing habits of college
students. She believes 45 percent of college students watch soap operas during the
afternoon. To further investigate, she selects a sample of 10.
1.3.1
1.3.2
1.3.3
Write down a probability distribution for the number of students in the sample who
watch soap operas.
[3]
Find the mean and variance of this distribution.
[4]
What is the probability of finding exactly four watch soap operas?
[3]
Question 2[30 marks]
2.1 Consider the experiment of tossing a fair coin three times.
2.1.1 Develop a tree diagram for the experiment.
[2]
2.1.2 List the experimental outcomes.
[8]
2.1.3 What is the probability for each experimental outcome?
[1]
2.2. A committee of 5 persons is to be formed from 6 men and 4 women. In how many
ways can this be done when:
2.2.1 at least 2 women are included?
[4]
2.2.2 at most 2 women are included?
[4)
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2.3
The following information is based on the trends in the United States published by
the Food Marketing Institute, Washington, D.C.The columns represent length of
customer loyalty (in years) at a primary supermarket. The rows represent regions in
United States
Region
East
Midwest
South
West
Column
total
Lessthan a
yel-ar
32
31
53
41
1-2
years
54
68
92
56
Loyalty
3-4
years
59
68
93
67
5-9
years
112
120
158
78
10-14
years
77
63
106
45
15 or more
years
118
173
158
86
Row
total
452
523
660
373
157
270
287
468
291
535 2008
What is the probability that a customer chosen at random
2.3.1 has been loyal at least 10 years or is from Midwest?
[2]
2.3.2 has been loyal at least 10 years, given that he or she is from South or West?
[3]
2.4.
In a certain assembly plant, three machines, B1,B2,and 83, make 30%, 45%, and 25%,
respectively, of the products. It is known from past experience that 2%, 3%, and 2% of the
products made by each machine, respectively, are defective. Now, suppose that a finished
product is randomly selected.
2.4.1. If a finished product is randomly selected and is found to be defective, what is the
probability that it was made by machine B2?
[4]
2.4.2. If a finished product is randomly selected and is found to be defective, what is the
probability that it was made by machines B1and B2?
[2]
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Question 3 [20 marks]
3.1. The probability distribution of X, the number of imperfections per 10 meters of a
synthetic fabric in continuous rolls of uniform width, is given by
X
0 1234
p(x) 0.41 0.37 0.16 0.05 0.01
3.1.1. What is the probability of no imperfections in 10 meters of a synthetic fabric? [lJ
3.1.2. What is the probability that there are at least three imperfections in 10 meters of a
synthetic fabric?
[2J
3.1.3. What is the expected number of imperfections in 10 meters of a synthetic fabric?
[3J
3.1.4. What is the standard deviation of the imperfections in 10 meters of a synthetic
fabric?
[3J
3.1.5. Compute the coefficient of variation for the imperfections in 10 meters of a
synthetic fabric?
[2J
3.1.6. Construct the cumulative distribution of X and hence find the median of X
[4J
3.2 A random variable X has a meanµ = 10 and a variance 0' 2 = 4. Use Chebyshev's theorem
to estimate P(IX - 101 2::3)
[SJ
Question 4 [25 marks]
4.1 Let X be random variable with a probability mass function given by
e-µµx
i f(x) = ~,for
0,
x = 0,1,2 ... ,
elsewhere
z:;: Show that
0 f(x) = 1
[SJ
4.2 The number of typing errors made by a typist has Poisson distribution with an average
of four errors per page. If more than four errors on a given page, the typist must retype
the whole page.
4.2.1 What is the probability that a certain page does not have to be retyped?
[SJ
4.2.2 What is the expected number of typing errors will the typist make in 5 pages? [2]
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4.2.3 What is the standard deviation number of typing errors will the typist make in 5 pages
[3]
4.3
University and college students average 7.2 hours of sleep per night, with a standard deviation
of 40 minutes. If the amount of sleep is normally distributed,
4.3.1 What proportion of university and college students sleep for more than 8 hours? [S]
4.3.2 Find the amount of sleep that is exceeded by only 25% of students.
[5]
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Standard Normal Probabilities
Table entry for z is the area under the standard normal curve
z
to the left of z.
z .oo .01
.02
.03
.04
.05
.06
.07
.08
.09
-3.4 .0003
-3.0 .0013
.0047
.0062
.0082
-2.2 .0139
-2.1
'
.0228
.0548
t:1,5 . .0668
-1.4 .0808
-1.2 .1151
~1.1 ".. 13i7
-1.0 .1587
.2119
.0003 .0003
:_Qo-6s.&9..Q5
.0006
.0013
.0003 .0003
.0006
~,...,-~-
.0012 .0012
.0045
.0069
.0080
.0136
.0044
.0059
.0078
.0132
.0023
~0032
.0043
.005:f
.0075
.0099
.0129
.0023
.0031
.0041
.0073
.0096
.0125
.0217
-,.. ,0274
.0351 .0344
,0212 .0207
.02.6.8....., .0262
.0336 .0329
.0003 .0003 .0003 .0003 .0002
.0004 .0004 .0003
.0006 .0006 .0005 .0005 .0005
.0008 .0007 ·_.:._q_gf)7
.ooTr .0011 .0011 .0011 .0010 .0010
'1f016
.001~014
· .0014
.0022 .0021 .0021
.0030 . .0029 .002lf
.0040 .0039 .0038 .0037
.0052 ;0051 .0049 .0048!
.0071 .0069 .0068 .0066 .0064
.0094 .0091 .0089 .0087 ,0084,
.0122 .0119 .0116 .0113 .0110
.0158 .0154 :0150 :0146 .014'.39
.0197
.0188 .0183
.0233
.0322 .0314
.040f , .
.0301
.0537 .0526 .0516 .0505 .0495
.0655 .0643=J2630
,.Jl§,18 .0606
.0793 .0778 .0764 .0749 .0735
.. 0951 .0934..
.1131 .1112 .1093 .1075 .1056
_.J33~-.-1314' .1292 -:1i7C· - .1251
.1562 .1539 .1515 .1492 .1469
.1788 - ".'°1762 :1735
.2061 .2033
.1977
.0485
.1038
.1230
.0475
.1020
.1423
.1922
.0465 .0455
.0571 .0559 -
.0985
.1401 .1379
.1635-· ,.1611•
.1894 .1867
-0.6 .2743 .2709
-0.2
cO,L
-0.0
.3409
2
.4207 .4168
.i§fil.-- '"'ff55[,
.5000 .4960
.2676 .2643
.3372 .3336
.4129 .4090
.4522 - _·.4483.
.4920 .4880
.2611 .2578 .2546
---· --'!>. ,19!2
-~§7?
.3300 .3264 .3228
.366~-· ~6_:g_~-.3594
.4052 .4013 .3974
~4443 .4404 · .4364
.4840 .4801 .4761
.2514
.2843
.3192
· .3557
.3936
.4325 ·
.4721
.2483 .2451
.28!.Q _.ill,6-,
.3156 .3121
.3520
.3897 .3859
.4286 .4247
.4681 .4641
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r.
Standard Normal Probabilities
z Table entry for is the area under the standard normal curve
z
to the left of z.
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0 .5000 .5040
t9,l .5398 .5438
0.2 .5793 .5832
'0.3 .6179 .6217
0.4 .6554 .6591
1().5 .6915 .6950
0.6 .7257 .7291
,0.7 .7580 ·.7611
0.8 .7881 .7910
:0-~ .8159 .8186
1.0 .8413 .8438
1.1 .8643 .8665
1.2 .8849 .8869
1.3 .9032 .9049
1.4 .9192 .9207
i.5 .9332 .9345
1.6 .9452 .9463
1.7 .9554 .9564
1.8 .9641 .9649
1.9 .9713 .9719
2.0 .9772 .9778
2.1 .9821 .9826
2.2 .9861 .9864
2.3 .9893 .9896
2.4 .9918 .9920
·2.5 .9938 .9940
2.6 .9953 .9955
2.7 .9965 .9966
2.8 .9974 .9975
2.9 .9981 .9982
3.0 .9987 .9987
L3.1 .9990 .9991
3.2 .9993 .9993
3.3 .9995 .9~95
3.4 .9997 .9997
.5080
.t478
.5871
.6255
.6628
.6985
.7324
.7642
.7939
.8212
.8461
.8686
.8888
.9066
.9222
.9357
.9474
.9$73
.9656
.~726
.9783
.9830
.9868
.9898
.9922
.9941
.9956
.9967
.9976
.9982
.9987
.9991
.9994
.9995
.9997
.5120
.5517
.5910
.6293
.6664
.7fo9
.7357
.7673
.7967
,8?38
.8485
.8708
.8907
.9082
.9236
.9370
.9484
.9582
.9664
_:97J2
.9788
.9834
.9871
.9901
.9925
.9943
.9957
.9968
.9977
.9983
.9988
.9991
.9994
.9996
.9997
.5160
.5557
.5948
.6331
.6700
.7054
.7389
.7704
.7995
.8264
.8508
.8729
.8925
.9099
.9251
.9382
.9495
.9591
.9671
.~7J~.
.9793
.9838
.9875
.9904
.9927
.9945
.9959
.9969
.9977
.9984
.9988
.9992
.9994
.9996
.9997
.5199
.5596
.5987
.63.68
.6736
.7088
.7422
.7734
.8023
.8289
.8531
.8749
.8944
.9115
.9265
.9394
.9505
.9599
.9678
.9744
.9798
.9842
.9878
.9906
.9929
.9946
.9960
.9970
.9978
.9984
.9989
.9992
.9994
.9996
.9997
.5239
.5636
.6026
.6406
.6772
.7123
.7454
.7764
.8051
.8315
.8554
.87i0
.8962
.9-131
.9279
.9406
.9515
.9608
.9686
.97~0
.9803
.9846
.9881
.9909
.9931
.9948
.9961
.9971
.9979
.9985
.9989
.9992
.9994
.9996
.9997
.5279 .5319
.5675 .5714
.6064 .6103
.6443 .6480.
.6808 .6844
.7157 .7190
.7486 .7517
.7794 . .7823
.8078 .8106
.8340 .8365
.8577 .8599
.8790 .8810
.8980 .8997
.9147 .9162
.9292 .9306
.9418 .9429
.9525 .9535
.9616 .. 9625
.9693 .9699
.~75§ .9761
.9808 .9812
.9850 .9854
.9884 .9887
.9911 .9913
.9932 .9934
.9949 .9951
.9962 .9963
.9972 .9973
.9979 .9980
.9985 .9986
.9989 .9990
.9992 .9993
.9995 .9995
.9996 .9996
.9997 .9997
.5359
.5753
.6141
.6517_,
.6879
.7224
.7549
.7852
.8133
.8389
.8621
.8830
.9015
.9117"
.9319
.944-1
.9545
.9633 -
.9706
.9767
.9817
.9857
.9890
.991§
.9936
.9952
.9964
.9974
.9981
.9986
.9990
.9993
.9995
.9997
.9998