EXAM – 2023
TRIMESTER 2
FRONT PAGE
EEEN 220
SIGNALS, SYSTEMS
& STATISTICS 1
25/10/2023
Time Allowed:
TWO HOURS
CLOSED BOOK
Permitted materials: Only silent non-programmable calculators or silent programmable calcula-
tors with their memories cleared are permitted in this examination.
A single double sided A4 sheet of formulae is permitted (which does not
need to be submitted).
Instructions:
Attempt ALL Questions.
NOT all questions have the same marks value.
There are 50 marks in total.
EEEN 220
Page 1 of 11
1. Sample statistics
[6 MARKS]
Measurements of signal strength y versus distance x for a given wireless channel were
made, giving the results below:
x (km)
5
22
18
9
33
21
y (dB)
24
15
16
9
7
13
(a) Calculate the sample means ¯x and ¯y.
[2 MARKS]
(b) Calculate the sample standard deviations sx and sy
[2 MARKS]
(c) Calculate the correlation coefficient between x and y.
[2 MARKS]
EEEN 220
Page 2 of 11
2. Conditional Probability
[10 MARKS]
In the process of producing engine valves, the valves are subjected to a first grind. Valves
having thickness within the specification are ready for installation. Valves having thick-
ness above the specification are reground, while those with thickness below the specifi-
cation are scrapped. After the first grind, 70% meet specification, 20% are reground, and
10% are scrapped. Of those valves that are reground, 90% meet specification, and 10%
are scrapped.
Hint: Let K denote the event that a valve meets the specification, R denote the event that
a valve is reground, and S the event that a valve is scrapped. Then P(K ∩ Rc) = 0.7,
P(R) = 0.2, P(S ∩ Rc) = 0.1, P(K|R) = 0.9, P(S|R) = 0.1.
(a) Find the probability that a valve is ground only once.
[1 MARK]
(b) Given that a valve is not reground, what is the probability that it is scrapped?
[1 MARK]
(c) Find the probability that a valve is scrapped.
[2 MARKS]
(d) Given that a valve is scrapped, what is the probability that it was ground twice?
[2 MARKS]
(e) Find the probability that the valve meets the specification (after the first or second
grind).
[2 MARKS]
(f) Given that a valve meets the specification, what is the probability that it was ground
twice?
[2 MARKS]
EEEN 220
Page 3 of 11
3. 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.
3
Y
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
(e) Determine E{X }
[2 MARKS]
(f) Determine the expectation E{XY }.
[3 MARKS]
(g) Compute the covariance between X and Y .
[2 MARKS]
EEEN 220
Page 4 of 11
4. 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]
EEEN 220
Page 5 of 11
5. Uncertainties
[5 MARKS]
We wish to measure a change between two independent measurements, expressed 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]
EEEN 220
Page 6 of 11
6. Maximum Likelihood
[7 MARKS]
Derive the maximum likelihood estimator for the location parameter µ and the scale pa-
rameter b from samples 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).
EEEN 220
Page 7 of 11
7. 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 1.2 calls per
minute. The mean call duration (assumed exponentially distributed) is 11.6 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 2% 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
Page 8 of 11
Tables for EEEN220
Normal Distribution
Z
x
1
Pr[X ≤ x] = Φ(x) = 0.5 +
√
e−y2/2 dy
0
2π
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
EEEN 220
Page 9 of 11
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
Page 10 of 11
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
Page 11 of 11
