MER301: Engineering
Reliability
LECTURE 7:
Chapter 3: 3.12-3.13
Functions of Random Variables, and the Central
Limit theorem
L Berkley Davis
Copyright 2009
MER035: Engineering Reliability
Lecture 7
1
Summary
 Functions of Random Variables
 Linear Combinations of Random Variables
 Non-Independent Random Variables
 Non-linear Functions of Random Variables
 Central Limit Theorem
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 7
2
Functions of Random Variables
 In most cases, the independent
variable Y will be a function of
multiple x’s
Y  fn( x1 , x2, ,...xi )
 The function may be linear or nonlinear, and the x variables may be
independent or not…
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 7
3
Functions of Random Variables
(3-27)
(3-28)
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 7
4
Exercise 7.1: Linear Combination of Independent
Random Variables-Tolerance Example




Assembly formed from parts
A,B,C
Mean and Std Dev for the
parts are 10mm/0.1mm for A,
2mm/0.05mm for both B and
C
Dimensions are independent
Find the Mean/Std Dev of the
gap D ,and find the
probability D is less than
5.9mm
3-42
3-181
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 7
5
Exercise 7.1: Linear Combination of Independent
Random Variables-Tolerance Example: Solution
 We have
D  A B C
 The Expected Value of D is given by
 D  E( D)  E( A)  E( B)  E(C)  10  2  2  6mm
 The Variance of D is
V ( D)   D2  0.12  0.052  0.052  0.015
 And the Standard Deviation is  D
 0.1225
 The probability D<5.9 is
P( D  5.9)  P( D  D ) /  D  (5.9  6) / 0.1225  PZ D  0.82
and
L Berkley Davis
Copyright 2009
P  0.2072
MER301: Engineering Reliability
Lecture 7
6
Non-Independent Random Variables
 In many cases, random variables
are not independent. Examples
include:
 Discharge pressure and temperature
from a gas turbine compressor
 Cycles to crack initiation of a metal
subjected to an alternating stress…
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 7
7
Non-Independent Random Variables
P3
18
16
14
10
8
T3
825
783
736
618
547
(3-29)
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 7
8
Independent Random Variables
Independent Variable Example
X1
X2
X1*X2
8.29
7.49
8.65
6.83
4.58
10.73
7.22
9.75
11.35
1.91
9.17
8.24
10.4
8.52
6.43
9.05
10.38
5.5
8.52
8.95
14
12
10
X2
8
6
4
2
0
0
2
8.27
8.94
7.91
8.66
9.29
8.44
7.99
11.79
9.44
9.28
6.85
Independent Variables X1 and X2
5.87
10.8
8.15
7.19
5.07
8.44
9.99
7.74
9.12
L Berkley Davis
Copyright 2009
4
6
X1
8
10
12
68.56
66.96
68.42
59.15
42.55
90.56
57.69
114.95
107.14
17.72
62.81
48.37
112.32
69.44
46.23
45.88
87.61
54.95
65.94
81.62
X1
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
X2
8.098
0.505719706
8.52
8.52
2.26164728
5.115048421
1.710429055
-1.118926452
9.44
1.91
11.35
161.96
20
8.4615
0.346079756
8.44
8.44
1.547715719
2.395423947
0.834027522
-0.120633666
6.72
5.07
11.79
169.23
20
X1*X2
68.4435
5.546376687
66.45
#N/A
24.8041506
615.2458871
0.052461383
0.306733797
97.23
17.72
114.95
1368.87
20
Correlation of X1 and X2
-0.023160446
(3-29)
Non-Independent Random Variables
% Area
Salt Conc
X*Y
0.19
3.8
0.722
0.15
0.57
0.4
0.7
0.67
0.63
0.47
0.75
0.6
0.78
0.81
0.78
0.69
1.3
1.05
1.52
1.06
1.74
1.62
5.9
14.1
10.4
14.6
14.5
15.1
11.9
15.5
9.3
15.6
20.8
14.6
16.6
25.6
20.9
29.9
19.6
31.3
32.7
0.885
8.037
4.16
10.22
9.715
9.513
5.593
11.625
5.58
12.168
16.848
11.388
11.454
33.28
21.945
45.448
20.776
54.462
52.974
Salt Conc vs % Area
35
% Area covered by Roads
30
25
20
15
10
5
0
0
L Berkley Davis
Copyright 2009
0.5
1
Salt Conc
1.5
2
%Area
%Area
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
0.824
0.09828369
0.725
0.78
0.43953803
0.19319368
-0.0119903
0.7007578
1.59
0.15
1.74
16.48
20
Correlation of
Salt and Area
0.9754034
Salt Conc
17.135
1.768038982
15.3
14.6
7.906910702
62.51923684
-0.112043442
0.54447337
28.9
3.8
32.7
342.7
20
X*Y
17.33965
3.666446048
11.421
#N/A
16.3968452
268.8565325
0.898247041
1.398094431
53.74
0.722
54.462
346.793
20
(3-29)
Functions of Non-Independent Random
Variables
   
2
Y
2
X1
L Berkley Davis
Copyright 2009
2
X2

2
X3

2
X4
cov( X 1 , X 2 )  cov( X 1 , X 3 )  cov( X 1 , X 4 ) 
2  


cov(
X
,
X
)

cov(
X
,
X
)

cov(
X
,
X
)
2
3
2
4
3
4 

Non-Linear Functions
 Physics dictate many engineering
phenomena are described by non-linear
equations.Examples include
 Heat Transfer
 Fluid Mechanics
 Material Properties
 Non-Linear Functions introduce the
concept of Propagation of Errors when
doing Probabilistic Analysis
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 7
12
Non-Linear Functions
(3-37)
(3-38)
 These equations are used in many
aspects of engineering, including
analysis of experimental data and the
analysis of transfer functions obtained
from Design of Experiment results
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 7
13
Analytical Prediction of Variance
 The Partial Derivative(Propagation of Errors) Method
can be used to estimate variation when an
analytical model of Y as a function of X’s is
available
Y  fn X , X , X ,........, X 
1
 Y 
2
2
 Y 
3
2
  X2  
  X2
 Y2  
 X 1 
 X 2 
1
2
n
 Y
         
 X N
2
 2
  X n

 This approach can be used to estimate the variance
for either physics based models or for empirical
models
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 7
14
Example 7.2: Non-Linear
 
   

4



  

Functions
P
I
R


  

2
P
2
I
R
3-44
3-32 and 3-33
  I2  R  202  80  32000watts
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 7
15
2
Random Samples,Statistics
and Central Limit Theorem
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 7
16
X  diameter
Impact of Measurement System Variation on
Variation in Experimental Data
P(0.2485  X  0.2515)  0.919
  0.2508
  0.0005

 actual
actual
Impact of Measurement System Variation on
Variation in Experimental Data
Impact of Measurement System Variation on
 actual
Variation in Experimental Data
Impact of Measurement System Variation on
 actual
Variation in Experimental
Data
 actual
Impact of Measurement System Variation on
 actual
 actual USL
LSL
Variation in Experimental
Data
 actual 2
2
 obs
 actual
  m System Variation on
Impact
of 
Measurement
 actual
 actual USL
LSL
Variation in Experimental
Data
 actualsystem
Product
2
 obs

 Measurement
 2
variance
= Product
Product
Mean
Impact
of
Measurement
=
Std.
Dev. actual variance m System Variation on
 actual
 actual USL Observed
Defects
LSL
Variation
in Experimental
Data
 actualsystem
Product
2t
m obsmeasuremen


 Measurement
 2
actImpact
 =actual
variance
Product
Mean
of
Measurement
= Product
Std.
Dev. actual variance m System Variation on

Actual
Defects
obs

observed
 actual
LSL
obs USL LSL
 actual USL Observed
Defects
Variation
in Experimental
Data
 actualsystem
Product
2t
m obsmeasuremen


 Measurement
 2
actImpact
 =actual
variance
Product
Mean
of
Measurement
= Product
Std.
Dev. actual variance m System Variation on

Actual
Defects
obs

observed
 actual
LSL
obs USL LSL
 actual USL Observed
Defects
Variation
in Experimental
Data
 actualsystem
Product
2t
m obsmeasuremen


 Measurement
 2
actImpact
 =actual
variance
Product
Mean
of
Measurement
= Product
Std.
Dev. actual variance m System Variation on

Actual
Defects
obs

observed
 actual
LSL
obs USL LSL
 actual USL Observed
Defects
Variation
in Experimental
Data
 actualsystem
Product
2t
m obsmeasuremen


 Measurement
 2
actImpact
 =actual
variance
Product
Mean
of
Measurement
= Product
Std.
Dev. actual variance m System Variation on

Actual
Defects
obs

observed
 actual
LSL
obs USL LSL
 actual USL Observed
Defects
Variation
in Experimental
Data
 actualsystem
Product
Measurement
2t
m obsmeasuremen

  actual
  m2
act  =actual
variance
variance
Product
Mean
Unio n Coll eg e
Mec ha nic al Engi ne eri ng
LSL
 actual
USL
MER30 1: Engi ne erin g R eliability
Le ct ur e 1 6
Unio n Coll eg e
Mec ha nic al Engi ne eri ng


obs

2
actual

2
m
MER30 1: Engi ne erin g R eliability
Le ct ur e 1 6
Unio n Coll eg e
Mec ha nic al Engi ne eri ng
MER30 1: Engi ne erin g R eliability
Le ct ur e 1 6
Unio n Coll eg e
Mec ha nic al Engi ne eri ng
P( X 1  and  X 2  ...  X 10 )  ==PProduct
( X 1 )Std.
 P( X )  ...  P( X 10 )
Product
MeanDev. 2
MER30 1: Engi ne erin g R eliability
Le ct ur e 1 6
 obs
LSL
Product
variance
Measurement system
variance
Observed Defects
LSL
Unio n Coll eg e
Mec ha nic al Engi ne eri ng
L Berkley Davis
Copyright 2009
 obs
USL
act  actualm  measuremen t
Actual Defects
obs  observed
= Product Std. Dev.
obs
 observed
Observed
Defects
USL LSL Actual
 actualDefects
USL
Unio n Coll eg e
Mec ha nic al Engi ne eri ng
LSL
 actual

m obsmeasuremen
t variance

  actual
m
act  =actual
variance
Product
Mean
= Product
Std.
Dev.
obs
 observed
USL LSL Actual
 actualDefects
Observed
Defects
USL

Unio n Coll eg e
Mec ha nic al Engi ne eri ng
2t
m obsmeasuremen

  actual

act  =actual
variance
variance
Product
Mean
= Product
Std.
Dev.
obs
 observed
Observed
Defects
actualsystem
Measurement
2
Product
MER30 1: Engi ne erin g R eliability
Le ct ur e 1 6
 obs
LSL
actualsystem
Measurement
2
2
Product
MER30 1: Engi ne erin g R eliability
Le ct ur e 1 6
 obs
m
USL LSL Actual
 actualDefects
USL
Unio n Coll eg e
Mec ha nic al Engi ne eri ng
 obs
LSL
2t
m obsmeasuremen

  actual

act  =actual
variance
variance
Product
Mean
= Product
Std.
Dev.
Actual Defects
obs
 observed
Observed
Defects
Product
MER30 1: Engi ne erin g R eliability
Le ct ur e 1 6
USL
Unio n Coll eg e
Mec ha nic al Engi ne eri ng
 obs
LSL
Unio n Coll eg e
Mec ha nic al Engi ne eri ng
LSL
Unio n Coll eg e
Mec ha nic al Engi ne eri ng
m
m  measuremen
act  =actual
variance
Product
MeanDev. t variance
= Product
Std.
Actual Defects
obs
 observed
Observed
Defects
Product
MER30 1: Engi ne erin g R eliability
Le ct ur e 1 6
USL
 obs
Measurement system
2
MER30 1: Engi ne erin g R eliability
Le ct ur e 1 6
USL
Measurement system
act  actualm  measuremen t
obs  observed
Actual Defects
MER30 1: Engi ne erin g R eliability
Le ct ur e 1 6
MER30 1: Engi ne erin g R eliability
Le ct ur e 1 6
MER301: Engineering Reliability
Lecture 6
17
Example 7.3: Throwing Dice..
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 7
18
Dice Example
Dice 1
3
3,1
3,2
3,3
3,4
3,5
3,6
1
2
4
5
6
1
1,1
2,1
4,1
5,1
6,1
2
1,2
2,2
4,2
5,2
6,2
Dice 2
3
1,3
2,3
4,3
5,3
6,3
4
1,4
2,4
4,4
5,4
6,4
5
1,5
2,5
4,5
5,5
6,5
6
1,6
2,6
4,6
5,6
6,6
 36 Elementary Outcomes

Probability of a specific outcome is 1/36

Probability of the event “sum of dice equals 7” is 6/36
 Addition-P(A or B)

Events “sum=7” and “sum=10” mutually exclusive P=(6/36+3/36)

Events “sum=7” and “dice 1=3” not mutually exclusive P=(11/36)
 Multiplication-P(A and B)

Events A=(6,6) and B=(6,6 repeated) are independent P=(1/36)2

Event A= (6,6) given B=(n1=n2=even) are not independent P=(1/3)
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 7
19
Central Limit Theorem
(3-39)
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 7
20
Central Limit Theorem
Impact of Measurement System Variation on
Variation in Experimental Data
 actual
 3
 actual
LSL
16 Sample Data Sets- mean =48, standard deviation= 3
 actual
  48
USL
Set 1
1
2
Expect
X  48
S 3obs
LSL
L Berkley Davis
Copyright 2009

45.93

2 48.73
42.93
actual
2

42.46
m
44.17
Set 4
51.83
Set 5
Set 6
Set 7
Set 8
Set 9
51.6
53.2
41.45
47.3
51.29
45.07
45.68
41.65
46.3
46.79
48.4
46.9
47.02
46.89
52.03
47.74
47.44
46.46
53.92
4
50.6
55.13
46.04
52.98
43.16
49.62
50.71
53.76
47.75
5
Product
46.43
50.03
46.86
Measurement
system 45.46
50.27
43.67
variance
43.44
46.91
47.9
varianceMean
= Product
Product
=
Std. Dev.
48.08
47.03
54.58
42.77
50.27 Observed
49.4
50.62
49.79
Defects
45.79
40.27
52.34
44.16
46.04
43.88
44.65
50.08
48.97
45.18
8
47.28
48.39
48.42
m 49.67
measuremen
t45.27
act  actual
46.09
45.23
51.33
44.4
51.91
48.34
48.01
49.36
Actual Defects
obs48.64
 observed
49.85
44.92
51.71
53.65
49.46
48.22
50.49
9
50.59
10
52.33
11
49.33
12
46.64
MER30 1: Engi ne
erin g R eliability
Le ct ur e 1 6
13
48.13
14
49.77
15
46.25
16
47.3
43.32
47.92
47.07
51.91
49.37
51.42
43.9
48.56
50.13
44.84
45.48
42.72
50.07
47.56
53.98
49.63
49.92
42.68
45.54
49.65
52.89
45.66
46.3
47.25
54.62
50.48
46.71
47.65
48.91
51.23
48.26
44.34
7
for a set of
sample data
Unio n Coll eg e
Mec ha nic al Engi ne eri ng
47.1
44.74
obs
Set 3
3
6
USL

Set 2
46.43
46.04
53.56
49.6
56.51
50.55
46.35
46.99
49.64
51.76
49.54
50.55
51.11
47.05
50.64
46.18
50.41
48.43
46.68
52
Central Limit Theorem
Impact of Measurement System Variation on
Variation in Experimental Data
 actual
 3
 actual
LSL
actual
48
1
USL
2

3
4
LSL
 obsSample
USL
Unio n Coll eg e
Mec ha nic al Engi ne eri ng
L Berkley Davis
Copyright 2009
Set 2
Set 3
Set 4
Set 5
Set 6
Set 7
Set 8
Set 9
47.1
44.17
  m2
45.93
48.73
51.83
51.6
53.2
41.45
47.3
51.29
42.93
42.46
45.07
45.68
41.65
46.3
46.79
46.9
47.02
46.89
52.03
47.74
47.44
46.46
53.92
46.04
52.98
43.16
49.62
50.71
53.76
47.75
46.863
Set
50.27
Set 4
43.67
Set 5
45.46
Set 6
43.44
Set
54.58
42.77
45.79
40.27
52.34

2
actual
44.74
48.4
Product
Measurement system
m  measuremen t
actual
5 act Set
46.43
50.03
1  observed
Set
2
Actual Defects
obs
6
7
8
9
10
11
Standard Deviation
12
Sample Variance
13
Kurtosis
14
Skewness
15
Range
16
Minimum
Maximum
Sum
Count
Set 1
varianceMean
variance
= Product
Product
=
Std. Dev.
50.6 Defects
55.13
Observed
MER30 1: Engi ne erin g R eliability
Le ct ur e 1 6
Mean
Standard Error
Median
Mode

obs
16 Sample Data Sets- mean =48, standard deviation= 3
48.08
47.03
7
46.91
Set
8
44.16
47.9
Set
9
46.04
50.27
49.4
50.62
49.79
43.88
44.65
50.08
48.3275
49.1856
48.3719
48.66 47.4775
47.7338
47.5613
47.28
48.39
49.67
48.42
45.27
53.65
49.46
0.50234
0.89032
0.70222
0.78343
0.81784
0.94005
0.9694
48.97
47.6231 45.18
48.8475
50.59
46.09
48.105
48.895 45.23
48.49 51.33
49.665 44.4
46.43 43.32
47.83 50.13
48.51
52.33
51.91
48.34
48.01
49.36
47.92
44.84
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
49.33
49.85
48.64
44.92
51.71
47.07
45.48
2.00934
3.56129
2.80888
3.13372
3.27134
3.76022
3.8776
46.64
46.43
50.55
49.54
46.18
51.91
42.72
4.03746 12.68281 7.88979 9.820227 10.70167 14.13923 15.0358
48.13
46.04
46.35
50.55
50.41
49.37
50.07
-0.41421 -0.246065 0.618507 -0.12713 -1.596104 -0.488753 -1.064871
49.77
53.56
46.99
51.11
48.43
51.42
47.56
0.261972 0.747306 0.283518 -0.836719 0.26528 -0.147822 -0.232884
46.25
49.6
49.64
47.05
46.68
43.9
53.98
7.59
12.34
11.65
10.52
8.87
13.38
12.53
47.3
56.51
51.76
50.64
52
48.56
49.63
44.74
44.17
42.93
42.46
43.16
40.27
41.45
52.33
56.51
54.58
52.98
52.03
53.65
53.98
773.24
786.97
773.95
778.56
759.64
763.74
760.98
49.92
47.08
42.6846.3
45.54
2.91185
49.65
8.47885
52.89
0.47499
45.66
0.662715
46.3
11.08
47.25
42.68
53.76
761.97
54.6248.08
50.48
#N/A
46.71
2.94839
47.65
8.69298
48.91
-0.341887
51.23
0.517647
48.26
10.28
44.34
44.34
54.62
781.56
16
16
16
16
16
16
16
16
16
48.22
0.72796 50.49
0.7371
Central Limit Theorem
Means
48.3275
49.1856
48.3719
48.66
47.4775
47.7338
47.5613
47.6231
48.8475
  3 / 16  0.75
 3
  48
L Berkley Davis
Copyright 2009
Mean of 16 Sample Means
Mean
Standard Error
48.1987
0.20844
Median
48.3275
Mode
#N/A
Standard Deviation
0.62533
Sample Variance
0.391035
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
-1.480405
0.272586
1.708125
47.4775
49.18563
433.7881
9
Example 7.4: Sampling
Text
Example 3-48
3-44
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 7
24
Importance of the Central Limit Theorem
 Most of the work done by engineers relies on
using experimentally derived data for material
property values(eg, ultimate strength , thermal
conductivity) and functional parameters (heat
transfer coefficients), as well as measurements
of actual system performance. These data are
acquired by drawing samples from the full
population. For all of these, the quantities of
interest can be represented by a mean value
and some measure of variance. The Central
Limit Theorem provides quantitative guidance
as to how much experimentation must be done
to estimate the true mean of a population.
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 7
25
Summary
 Functions of Random Variables
 Linear Combinations of Random Variables
 Non-Independent Random Variables
 Non-linear Functions of Random Variables
 Central Limit Theorem
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 7
26