CLASS TEST 2 – 2022
TRIMESTER 2
EEEN 220
SIGNALS, SYSTEMS
& STATISTICS 1
Time Allowed:
FIFTY MINUTES
CLOSED BOOK
Permitted materials: Calculators. A single A4 sheet of formulae.
Instructions:
Attempt ALL Questions.
NOT all questions have the same marks value.
There are 44 marks in total.
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1. Conditional Probability
(10 marks)
Items are inspected for flaws by two quality inspectors. If a flaw is present, it
will be detected by the first inspector with probability 0.9, and by the second
inspector with probability 0.7. Assume the inspectors function independently.
(a) If an item has a flaw, what is the probability that it is found by both inspec-
tors?
(1 mark)
(b) If an item has a flaw, what is the probability that it is found by at least one
inspector?
(1 mark)
(c) If the second inspector were to only examine those items that have been
passed by the first inspector, what is the probability, for an item that has a
flaw, that the second inspector will find it?
(1 mark)
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(d) If the probability that an item has a flaw is 0.1, and an item is passed by the
first inspector, what is the probability that it actually has a flaw?
(3 marks)
(e) If the probability that an item has a flaw is 0.1, and an item is passed by the
both inspectors, what is the probability that it actually has a flaw?
(4 marks)
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2. Joint Probability Density
(12 marks)
Consider the density fX,Y(x, y) shown below. The shaded area has a constant
density, c. Elsewhere the density is zero. As always: explain all answers. Only
providing answers results in no marks.
Y
3
2
1
X
1
2
3
(a) Compute the value c of the density, where the density is not zero.
(1 mark)
(b) Determine the density fX(x) (a careful plot is ok).
(1 mark)
(c) Determine the function fX,Y( 1, y) (a careful plot is ok).
(1 mark)
2
(d) Determine the density fY|X(y|1) (a careful plot is ok).
(2 marks)
2
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(e) Determine E{X}
(2 marks)
(f) Determine the expectation E{XY}.
(3 marks)
(g) Compute the covariance between X and Y.
(2 marks)
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3. Distributions
(6 marks)
Derive an expression for the mean and variance of a continuous random variable
X which is uniformly distributed on the interval [a, b].
(Note: There are no marks for the answers, only for the working).
(a) Mean:
(2 marks)
(b) Variance:
(4 marks)
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4. Uncertainties
(5 marks)
We wish to measure a change between two independent measurements, ex-
pressed as a proportion of the initial value i.e., y = (x2 − x1)/x1.
(a) If x1 = 1.11 ± 0.01 and x2 = 0.99 ± 0.01, what is the value of y?
(1 mark)
(b) What is the uncertainty in y?
(4 marks)
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5. Maximum Likelihood
(7 marks)
Derive the maximum likelihood estimator for the location parameter
µ and the
scale parameter b of the Laplace distribution (which has p.d.f fX(x) = 1 e− |x−
µ|
b
).
2b
(Note: Most of the marks for this question are for careful explanation of what
you are doing and the reasons for it).
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6. Traffic
(4 marks)
A car insurance company is setting up a 0800 number and call centre for new
customers. They estimate that during the busy hour the (Poisson) call arrival
rate is 3 calls per minute. The mean call duration (assumed exponentially dis-
tributed) is 7 minutes. The company assumes that customers who encounter
congestion on calling will immediately call their competitor, and don’t want this
to happen for any more than 5% of callers.
(a) What is the busy hour traffic load? (provide the value and the units)
(2 marks)
(b) How many operators does the call centre require to meet the availability
target?
(2 marks)
EEEN 220
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Tables for EEEN220
Normal Distribution
Z
x
1
Pr[X ≤ x] = Φ(x) = 0.5 +
√
e−y2/2 dy
0
2
π
The table shows Φ(z) − 0.5 as a function of z > 0.
z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.0
.0000
.0040
.0080
.0120
.0160
.0199
.0239
.0279
.0319
.0359
0.1
.0398
.0438
.0478
.0517
.0557
.0596
.0636
.0675
.0714
.0753
0.2
.0793
.0832
.0871
.0910
.0948
.0987
.1026
.1064
.1103
.1141
0.3
.1179
.1217
.1255
.1293
.1331
.1368
.1406
.1443
.1480
.1517
0.4
.1554
.1591
.1628
.1664
.1700
.1736
.1772
.1808
.1844
.1879
0.5
.1915
.1950
.1985
.2019
.2054
.2088
.2123
.2157
.2190
.2224
0.6
.2257
.2291
.2324
.2357
.2389
.2422
.2454
.2486
.2517
.2549
0.7
.2580
.2611
.2642
.2673
.2704
.2734
.2764
.2794
.2823
.2852
0.8
.2881
.2910
.2939
.2967
.2995
.3023
.3051
.3078
.3106
.3133
0.9
.3159
.3186
.3212
.3238
.3264
.3289
.3315
.3340
.3365
.3389
1.0
.3413
.3438
.3461
.3485
.3508
.3531
.3554
.3577
.3599
.3621
1.1
.3643
.3665
.3686
.3708
.3729
.3749
.3770
.3790
.3810
.3830
1.2
.3849
.3869
.3888
.3907
.3925
.3944
.3962
.3980
.3997
.4015
1.3
.4032
.4049
.4066
.4082
.4099
.4115
.4131
.4147
.4162
.4177
1.4
.4192
.4207
.4222
.4236
.4251
.4265
.4279
.4292
.4306
.4319
1.5
.4332
.4345
.4357
.4370
.4382
.4394
.4406
.4418
.4429
.4441
1.6
.4452
.4463
.4474
.4484
.4495
.4505
.4515
.4525
.4535
.4545
1.7
.4554
.4564
.4573
.4582
.4591
.4599
.4608
.4616
.4625
.4633
1.8
.4641
.4649
.4656
.4664
.4671
.4678
.4686
.4693
.4699
.4706
1.9
.4713
.4719
.4726
.4732
.4738
.4744
.4750
.4756
.4761
.4767
2.0
.4772
.4778
.4783
.4788
.4793
.4798
.4803
.4808
.4812
.4817
2.1
.4821
.4826
.4830
.4834
.4838
.4842
.4846
.4850
.4854
.4857
2.2
.4861
.4864
.4868
.4871
.4875
.4878
.4881
.4884
.4887
.4890
2.3
.4893
.4896
.4898
.4901
.4904
.4906
.4909
.4911
.4913
.4916
2.4
.4918
.4920
.4922
.4925
.4927
.4929
.4931
.4932
.4934
.4936
2.5
.4938
.4940
.4941
.4943
.4945
.4946
.4948
.4949
.4951
.4952
2.6
.4953
.4955
.4956
.4957
.4959
.4960
.4961
.4962
.4963
.4964
2.7
.4965
.4966
.4967
.4968
.4969
.4970
.4971
.4972
.4973
.4974
2.8
.4974
.4975
.4976
.4977
.4977
.4978
.4979
.4979
.4980
.4981
2.9
.4981
.4982
.4982
.4983
.4984
.4984
.4985
.4985
.4986
.4986
3.0
.4987
.4987
.4987
.4988
.4988
.4989
.4989
.4989
.4990
.4990
3.1
.4990
.4991
.4991
.4991
.4992
.4992
.4992
.4992
.4993
.4993
3.2
.4993
.4993
.4994
.4994
.4994
.4994
.4994
.4995
.4995
.4995
3.3
.4995
.4995
.4995
.4996
.4996
.4996
.4996
.4996
.4996
.4997
3.4
.4997
.4997
.4997
.4997
.4997
.4997
.4997
.4997
.4997
.4998
3.5
.4998
.4998
.4998
.4998
.4998
.4998
.4998
.4998
.4998
.4998
3.6
.4998
.4998
.4999
.4999
.4999
.4999
.4999
.4999
.4999
.4999
3.7
.4999
.4999
.4999
.4999
.4999
.4999
.4999
.4999
.4999
.4999
3.8
.4999
.4999
.4999
.4999
.4999
.4999
.4999
.4999
.4999
.4999
3.9
.5000
.5000
.5000
.5000
.5000
.5000
.5000
.5000
.5000
.5000
4.0
.5000
.5000
.5000
.5000
.5000
.5000
.5000
.5000
.5000
.5000
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Erlang B Traffic Table
Pr[B]
N
0.0001
0.0005
0.001
0.005
0.01
0.02
0.05
0.1
0.15
0.2
0.30
0.40
1
.0001
.0005
.0010
.0050
.0101
.0204
.0526
.1111
.1765
.2500
.4286
.6667
2
.0142
.0321
.0458
.1054
.1526
.2235
.3813
.5954
.7962
1.000
1.449
2.000
3
.0868
.1517
.1938
.3490
.4555
.6022
.8994
1.271
1.603
1.930
2.633
3.480
4
.2347
.3624
.4393
.7012
.8694
1.092
1.525
2.045
2.501
2.945
3.891
5.021
5
.4520
.6486
.7621
1.132
1.361
1.657
2.219
2.881
3.454
4.010
5.189
6.596
6
.7282
.9957
1.146
1.622
1.909
2.276
2.960
3.758
4.445
5.109
6.514
8.191
7
1.054
1.392
1.579
2.158
2.501
2.935
3.738
4.666
5.461
6.230
7.856
9.800
8
1.422
1.830
2.051
2.730
3.128
3.627
4.543
5.597
6.498
7.369
9.213
11.42
9
1.826
2.302
2.558
3.333
3.783
4.345
5.370
6.546
7.551
8.522
10.58
13.05
10
2.260
2.803
3.092
3.961
4.461
5.084
6.216
7.511
8.616
9.685
11.95
14.68
11
2.722
3.329
3.651
4.610
5.160
5.842
7.076
8.487
9.691
10.86
13.33
16.31
12
3.207
3.878
4.231
5.279
5.876
6.615
7.950
9.474
10.78
12.04
14.72
17.95
13
3.713
4.447
4.831
5.964
6.607
7.402
8.835
10.47
11.87
13.22
16.11
19.60
14
4.239
5.032
5.446
6.663
7.352
8.200
9.730
11.47
12.97
14.41
17.50
21.24
15
4.781
5.634
6.077
7.376
8.108
9.010
10.63
12.48
14.07
15.61
18.90
22.89
16
5.339
6.250
6.722
8.100
8.875
9.828
11.54
13.50
15.18
16.81
20.30
24.54
17
5.911
6.878
7.378
8.834
9.652
10.66
12.46
14.52
16.29
18.01
21.70
26.19
18
6.496
7.519
8.046
9.578
10.44
11.49
13.39
15.55
17.41
19.22
23.10
27.84
19
7.093
8.170
8.724
10.33
11.23
12.33
14.32
16.58
18.53
20.42
24.51
29.50
20
7.701
8.831
9.412
11.09
12.03
13.18
15.25
17.61
19.65
21.64
25.92
31.15
21
8.319
9.501
10.11
11.86
12.84
14.04
16.19
18.65
20.77
22.85
27.33
32.81
22
8.946
10.18
10.81
12.64
13.65
14.90
17.13
19.69
21.90
24.06
28.74
34.46
23
9.583
10.87
11.52
13.42
14.47
15.76
18.08
20.74
23.03
25.28
30.15
36.12
24
10.23
11.56
12.24
14.20
15.30
16.63
19.03
21.78
24.16
26.50
31.56
37.78
25
10.88
12.26
12.97
15.00
16.13
17.51
19.99
22.83
25.30
27.72
32.97
39.44
26
11.54
12.97
13.70
15.80
16.96
18.38
20.94
23.89
26.43
28.94
34.39
41.10
27
12.21
13.69
14.44
16.60
17.80
19.27
21.90
24.94
27.57
30.16
35.80
42.76
28
12.88
14.41
15.18
17.41
18.64
20.15
22.87
26.00
28.71
31.39
37.21
44.41
29
13.56
15.13
15.93
18.22
19.49
21.04
23.83
27.05
29.85
32.61
38.63
46.07
30
14.25
15.86
16.68
19.03
20.34
21.93
24.80
28.11
31.00
33.84
40.05
47.74
40
21.37
23.41
24.44
27.38
29.01
31.00
34.60
38.79
42.48
46.15
54.24
64.35
50
28.87
31.29
32.51
35.98
37.90
40.26
44.53
49.56
54.03
58.51
68.46
80.99
60
36.62
39.40
40.80
44.76
46.95
49.64
54.57
60.40
65.63
70.90
82.70
97.63
70
44.58
47.68
49.24
53.66
56.11
59.13
64.67
71.29
77.26
83.32
96.95
114.3
80
52.69
56.10
57.81
62.67
65.36
68.69
74.82
82.20
88.91
95.75
111.2
130.9
90
60.92
64.63
66.48
71.76
74.68
78.31
85.01
93.15
100.6
108.2
125.5
147.6
100
69.27
73.25
75.24
80.91
84.06
87.97
95.24
104.1
112.3
120.6
139.7
164.3
EEEN 220
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12
Erlang C Traffic Table
Pr[W]
N
0.0001
0.0005
0.001
0.005
0.01
0.02
0.05
0.1
0.15
0.2
0.30
0.40
1
.0001
.0005
.0010
.0050
.0100
.0200
.0500
.1000
.1500
.2000
.3000
.4000
2
.0142
.0319
.0452
.1025
.1465
.2103
.3422
.5000
.6278
.7403
.9390
1.117
3
.0860
.1490
.1894
.3339
.4291
.5545
.7876
1.040
1.231
1.393
1.667
1.903
4
.2310
.3533
.4257
.6641
.8100
.9939
1.319
1.653
1.899
2.102
2.440
2.725
5
.4428
.6289
.7342
1.065
1.259
1.497
1.905
2.313
2.607
2.847
3.241
3.569
6
.7110
.9616
1.099
1.519
1.758
2.047
2.532
3.007
3.344
3.617
4.062
4.428
7
1.026
1.341
1.510
2.014
2.297
2.633
3.188
3.725
4.103
4.406
4.897
5.298
8
1.382
1.758
1.958
2.543
2.866
3.246
3.869
4.463
4.878
5.210
5.744
6.178
9
1.771
2.208
2.436
3.100
3.460
3.883
4.569
5.218
5.668
6.027
6.600
7.065
10
2.189
2.685
2.942
3.679
4.077
4.540
5.285
5.986
6.469
6.853
7.465
7.959
11
2.634
3.186
3.470
4.279
4.712
5.213
6.015
6.765
7.280
7.688
8.336
8.857
12
3.100
3.708
4.018
4.896
5.363
5.901
6.758
7.554
8.099
8.530
9.212
9.761
13
3.587
4.248
4.584
5.529
6.028
6.602
7.511
8.352
8.926
9.379
10.09
10.67
14
4.092
4.805
5.166
6.175
6.705
7.313
8.273
9.158
9.760
10.23
10.98
11.58
15
4.614
5.377
5.762
6.833
7.394
8.035
9.044
9.970
10.60
11.09
11.87
12.49
16
5.150
5.962
6.371
7.502
8.093
8.766
9.822
10.79
11.44
11.96
12.77
13.41
17
5.699
6.560
6.991
8.182
8.801
9.505
10.61
11.61
12.29
12.83
13.66
14.33
18
6.261
7.169
7.622
8.871
9.518
10.25
11.40
12.44
13.15
13.70
14.56
15.25
19
6.835
7.788
8.263
9.568
10.24
11.01
12.20
13.28
14.01
14.58
15.47
16.18
20
7.419
8.417
8.914
10.27
10.97
11.77
13.00
14.12
14.87
15.45
16.37
17.10
21
8.013
9.055
9.572
10.99
11.71
12.53
13.81
14.96
15.73
16.34
17.28
18.03
22
8.616
9.702
10.24
11.70
12.46
13.30
14.62
15.81
16.60
17.22
18.19
18.96
23
9.228
10.36
10.91
12.43
13.21
14.08
15.43
16.65
17.47
18.11
19.10
19.89
24
9.848
11.02
11.59
13.16
13.96
14.86
16.25
17.51
18.35
19.00
20.02
20.82
25
10.48
11.69
12.28
13.90
14.72
15.65
17.08
18.36
19.22
19.89
20.93
21.76
26
11.11
12.36
12.97
14.64
15.49
16.44
17.91
19.22
20.10
20.79
21.85
22.69
27
11.75
13.04
13.67
15.38
16.26
17.23
18.74
20.08
20.98
21.68
22.77
23.63
28
12.40
13.73
14.38
16.14
17.03
18.03
19.57
20.95
21.87
22.58
23.69
24.57
29
13.05
14.42
15.09
16.89
17.81
18.83
20.41
21.82
22.75
23.48
24.61
25.50
30
13.71
15.12
15.80
17.65
18.59
19.64
21.25
22.68
23.64
24.38
25.54
26.44
40
20.58
22.33
23.17
25.44
26.58
27.84
29.77
31.48
32.61
33.48
34.83
35.89
50
27.80
29.86
30.86
33.49
34.80
36.26
38.47
40.42
41.70
42.69
44.21
45.40
60
35.30
37.63
38.76
41.73
43.20
44.83
47.29
49.46
50.88
51.97
53.65
54.96
70
42.99
45.58
46.83
50.10
51.73
53.52
56.21
58.57
60.12
61.31
63.14
64.56
80
50.84
53.68
55.03
58.60
60.36
62.30
65.21
67.75
69.42
70.70
72.66
74.18
90
58.82
61.88
63.34
67.18
69.07
71.15
74.26
76.98
78.76
80.12
82.21
83.83
100
66.91
70.19
71.75
75.84
77.85
80.06
83.37
86.25
88.13
89.58
91.78
93.49
* * * * * * * * * * * * * * *
EEEN 220
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