L Berkley Davis Copyright 2009 MER035: Engineering Reliability Lecture 7 1 MER301: Engineering...
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L Berkley DavisCopyright 2009
MER035: Engineering ReliabilityLecture 7
1
MER301: Engineering Reliability
LECTURE 7:
Chapter 3: 3.12-3.13Functions of Random Variables, and the Central Limit theorem
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
2
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 DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
3
Functions of Random Variables
In most cases, the independent variable Y will be a function of multiple x’s
The function may be linear or non-linear, and the x variables may be independent or not…
),...,( ,21 ixxxfnY
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
4
Functions of Random Variables
(3-28)
(3-27)
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
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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
3-423-181
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
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Exercise 7.1: Linear Combination of Independent Random Variables-Tolerance Example: Solution
We have The Expected Value of D is given by The Variance of D is
And the Standard Deviation is The probability D<5.9 is
and
015.005.005.01.0)( 2222 DDV 1225.0D
mmCEBEAEDED 62210)()()()(
82.01225.0/)69.5(/)()9.5( DDD ZPDPDP
2072.0P
CBAD
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
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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 DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
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Non-Independent Random Variables
P3 T318 82516 78314 73610 6188 547
(3-29)
L Berkley DavisCopyright 2009
Independent Random Variables Independent Variable Example
X1 X2 X1*X28.29 8.27 68.567.49 8.94 66.968.65 7.91 68.426.83 8.66 59.154.58 9.29 42.55
10.73 8.44 90.567.22 7.99 57.699.75 11.79 114.95
11.35 9.44 107.141.91 9.28 17.729.17 6.85 62.818.24 5.87 48.3710.4 10.8 112.328.52 8.15 69.446.43 7.19 46.239.05 5.07 45.88
10.38 8.44 87.615.5 9.99 54.95
8.52 7.74 65.948.95 9.12 81.62
Independent Variables X1 and X2
0
2
4
6
8
10
12
14
0 2 4 6 8 10 12
X1
X2
X1 X2 X1*X2
Mean 8.098 8.4615 68.4435Standard Error 0.505719706 0.346079756 5.546376687Median 8.52 8.44 66.45Mode 8.52 8.44 #N/AStandard Deviation 2.26164728 1.547715719 24.8041506
Sample Variance 5.115048421 2.395423947 615.2458871Kurtosis 1.710429055 0.834027522 0.052461383Skewness -1.118926452 -0.120633666 0.306733797Range 9.44 6.72 97.23Minimum 1.91 5.07 17.72Maximum 11.35 11.79 114.95Sum 161.96 169.23 1368.87Count 20 20 20
Correlation of X1 and X2-0.023160446
(3-29)
L Berkley DavisCopyright 2009
%Area Salt Conc X*Y0.19 3.8 0.722
0.15 5.9 0.8850.57 14.1 8.0370.4 10.4 4.160.7 14.6 10.22
0.67 14.5 9.7150.63 15.1 9.5130.47 11.9 5.5930.75 15.5 11.6250.6 9.3 5.58
0.78 15.6 12.1680.81 20.8 16.8480.78 14.6 11.3880.69 16.6 11.4541.3 25.6 33.28
1.05 20.9 21.9451.52 29.9 45.4481.06 19.6 20.7761.74 31.3 54.4621.62 32.7 52.974
Non-Independent Random Variables
%Area %Area Salt Conc X*Y
Mean 0.824 17.135 17.33965Standard Error 0.09828369 1.768038982 3.666446048Median 0.725 15.3 11.421Mode 0.78 14.6 #N/AStandard Deviation 0.43953803 7.906910702 16.3968452Sample Variance 0.19319368 62.51923684 268.8565325Kurtosis -0.0119903 -0.112043442 0.898247041Skewness 0.7007578 0.54447337 1.398094431Range 1.59 28.9 53.74Minimum 0.15 3.8 0.722Maximum 1.74 32.7 54.462Sum 16.48 342.7 346.793Count 20 20 20
Salt Conc vs % Area
0
5
10
15
20
25
30
35
0 0.5 1 1.5 2
Salt Conc
% A
rea
cove
red
by
Ro
ads
Correlation ofSalt and Area
0.9754034(3-29)
L Berkley DavisCopyright 2009
Functions of Non-Independent Random Variables
),cov(),cov(),cov(
),cov(),cov(),cov(2
434232
413121
XXXXXX
XXXXXX 22222
4321 XXXXY
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
12
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 DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
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Non-Linear Functions
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
(3-37)
(3-38)
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
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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
This approach can be used to estimate the variance for either physics based models or for empirical models
nXXXXfnY ,........,,, 321
2
2
2
2
2
2
2
1
221 nX
NXXY X
Y
X
Y
X
Y
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
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Example 7.2: Non-Linear Functions
wattsRI 32000802022
222
4
RIPRIP
3-32 and 3-33
3-44
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
16
Random Samples,Statistics and Central Limit Theorem
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 6
17
Union Coll egeMechanical Engi neeri ng
MER301: Engi neering ReliabilityLecture 16
Actual Defects
LSL USL
= Product Std. Dev.= Product Mean
LSL USL
Observed Defects
Measurement system variance
Product variance
Impact of Measurement System Variation on Variation in Experimental Data
22mactualobs
actual
actual
actual
obs observedobs
tmeasuremenm
actualact
Union Coll egeMechanical Engi neeri ng
MER301: Engi neering ReliabilityLecture 16
Actual Defects
LSL USL
= Product Std. Dev.= Product Mean
LSL USL
Observed Defects
Measurement system variance
Product variance
Impact of Measurement System Variation on Variation in Experimental Data
22mactualobs
actual
actual
actual
obs observedobs
tmeasuremenm
actualact
Union Coll egeMechanical Engi neeri ng
MER301: Engi neering ReliabilityLecture 16
Actual Defects
LSL USL
= Product Std. Dev.= Product Mean
LSL USL
Observed Defects
Measurement system variance
Product variance
Impact of Measurement System Variation on Variation in Experimental Data
22mactualobs
actual
actual
actual
obs observedobs
tmeasuremenm
actualact
Union Coll egeMechanical Engi neeri ng
MER301: Engi neering ReliabilityLecture 16
Actual Defects
LSL USL
= Product Std. Dev.= Product Mean
LSL USL
Observed Defects
Measurement system variance
Product variance
Impact of Measurement System Variation on Variation in Experimental Data
22mactualobs
actual
actual
actual
obs observedobs
tmeasuremenm
actualact
Union Coll egeMechanical Engi neeri ng
MER301: Engi neering ReliabilityLecture 16
Actual Defects
LSL USL
= Product Std. Dev.= Product Mean
LSL USL
Observed Defects
Measurement system variance
Product variance
Impact of Measurement System Variation on Variation in Experimental Data
22mactualobs
actual
actual
actual
obs observedobs
tmeasuremenm
actualact
Union Coll egeMechanical Engi neeri ng
MER301: Engi neering ReliabilityLecture 16
Actual Defects
LSL USL
= Product Std. Dev.= Product Mean
LSL USL
Observed Defects
Measurement system variance
Product variance
Impact of Measurement System Variation on Variation in Experimental Data
22mactualobs
actual
actual
actual
obs observedobs
tmeasuremenm
actualact
Union Coll egeMechanical Engi neeri ng
MER301: Engi neering ReliabilityLecture 16
Actual Defects
LSL USL
= Product Std. Dev.= Product Mean
LSL USL
Observed Defects
Measurement system variance
Product variance
Impact of Measurement System Variation on Variation in Experimental Data
22mactualobs
actual
actual
actual
obs observedobs
tmeasuremenm
actualact
Union Coll egeMechanical Engi neeri ng
MER301: Engi neering ReliabilityLecture 16
Actual Defects
LSL USL
= Product Std. Dev.= Product Mean
LSL USL
Observed Defects
Measurement system variance
Product variance
Impact of Measurement System Variation on Variation in Experimental Data
22mactualobs
actual
actual
actual
obs observedobs
tmeasuremenm
actualact
Union Coll egeMechanical Engi neeri ng
MER301: Engi neering ReliabilityLecture 16
Actual Defects
LSL USL
= Product Std. Dev.= Product Mean
LSL USL
Observed Defects
Measurement system variance
Product variance
Impact of Measurement System Variation on Variation in Experimental Data
22mactualobs
actual
actual
actual
obs observedobs
tmeasuremenm
actualact
Union Coll egeMechanical Engi neeri ng
MER301: Engi neering ReliabilityLecture 16
Actual Defects
LSL USL
= Product Std. Dev.= Product Mean
LSL USL
Observed Defects
Measurement system variance
Product variance
Impact of Measurement System Variation on Variation in Experimental Data
22mactualobs
actual
actual
actual
obs observedobs
tmeasuremenm
actualact
)(...)()()...( 10211021 XPXPXPXXandXP
diameterX
Unio n Col l eg eMec ha nic al E ngi ne eri ng
ME R30 1: E ngi ne erin g R el iabi l i tyLe ct ur e 1 6
Actual Defects
LSL USL
= Product Std. Dev.= Product Mean
LSL USL
Observed Defects
Measurement system variance
Product variance
Impact of Measurement System Variation on Variation in Experimental Data
22mactualobs
actual
actual
actual
obs observedobs
tmeasuremenm
actualact
919.0)2515.02485.0( XP
0005.0
2508.0
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
18
Example 7.3: Throwing Dice..
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
19
Dice Example
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)
Dice 11 2 3 4 5 6
1 1,1 2,1 3,1 4,1 5,1 6,12 1,2 2,2 3,2 4,2 5,2 6,2
Dice 2 3 1,3 2,3 3,3 4,3 5,3 6,34 1,4 2,4 3,4 4,4 5,4 6,45 1,5 2,5 3,5 4,5 5,5 6,56 1,6 2,6 3,6 4,6 5,6 6,6
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
20
Central Limit Theorem
(3-39)
L Berkley DavisCopyright 2009
Union Coll egeMechanical Engi neeri ng
MER301: Engi neering ReliabilityLecture 16
Actual Defects
LSL USL
= Product Std. Dev.= Product Mean
LSL USL
Observed Defects
Measurement system variance
Product variance
Impact of Measurement System Variation on Variation in Experimental Data
22mactualobs
actual
actual
actual
obs observedobs
tmeasuremenm
actualact
Central Limit Theorem
48
3
16 Sample Data Sets- mean =48, standard deviation= 3
Set 1 Set 2 Set 3 Set 4 Set 5 Set 6 Set 7 Set 8 Set 9
1 47.1 44.17 48.73 51.83 51.6 53.2 41.45 47.3 51.29
2 44.74 45.93 42.93 42.46 45.07 45.68 41.65 46.3 46.79
3 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 46.43 50.03 46.86 50.27 43.67 45.46 43.44 46.91 47.9
6 48.08 47.03 54.58 42.77 45.79 40.27 52.34 44.16 46.04
7 50.27 49.4 50.62 49.79 43.88 44.65 50.08 48.97 45.18
8 47.28 48.39 49.67 48.42 45.27 53.65 49.46 48.22 50.49
9 50.59 46.09 45.23 51.33 44.4 43.32 50.13 49.92 54.6210 52.33 51.91 48.34 48.01 49.36 47.92 44.84 42.68 50.4811 49.33 49.85 48.64 44.92 51.71 47.07 45.48 45.54 46.7112 46.64 46.43 50.55 49.54 46.18 51.91 42.72 49.65 47.6513 48.13 46.04 46.35 50.55 50.41 49.37 50.07 52.89 48.9114 49.77 53.56 46.99 51.11 48.43 51.42 47.56 45.66 51.2315 46.25 49.6 49.64 47.05 46.68 43.9 53.98 46.3 48.2616 47.3 56.51 51.76 50.64 52 48.56 49.63 47.25 44.34
Expect
for a set of sample data
48X
3S
L Berkley DavisCopyright 2009
Union Coll egeMechanical Engi neeri ng
MER301: Engi neering ReliabilityLecture 16
Actual Defects
LSL USL
= Product Std. Dev.= Product Mean
LSL USL
Observed Defects
Measurement system variance
Product variance
Impact of Measurement System Variation on Variation in Experimental Data
22mactualobs
actual
actual
actual
obs observedobs
tmeasuremenm
actualact
Central Limit Theorem
48
316 Sample Data Sets- mean =48, standard deviation= 3
Set 1 Set 2 Set 3 Set 4 Set 5 Set 6 Set 7 Set 8 Set 9
1 47.1 44.17 48.73 51.83 51.6 53.2 41.45 47.3 51.29
2 44.74 45.93 42.93 42.46 45.07 45.68 41.65 46.3 46.79
3 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 46.43 50.03 46.86 50.27 43.67 45.46 43.44 46.91 47.9
6 48.08 47.03 54.58 42.77 45.79 40.27 52.34 44.16 46.04
7 50.27 49.4 50.62 49.79 43.88 44.65 50.08 48.97 45.18
8 47.28 48.39 49.67 48.42 45.27 53.65 49.46 48.22 50.49
9 50.59 46.09 45.23 51.33 44.4 43.32 50.13 49.92 54.6210 52.33 51.91 48.34 48.01 49.36 47.92 44.84 42.68 50.4811 49.33 49.85 48.64 44.92 51.71 47.07 45.48 45.54 46.7112 46.64 46.43 50.55 49.54 46.18 51.91 42.72 49.65 47.6513 48.13 46.04 46.35 50.55 50.41 49.37 50.07 52.89 48.9114 49.77 53.56 46.99 51.11 48.43 51.42 47.56 45.66 51.2315 46.25 49.6 49.64 47.05 46.68 43.9 53.98 46.3 48.2616 47.3 56.51 51.76 50.64 52 48.56 49.63 47.25 44.34
Sample Set 1 Set 2 Set 3 Set 4 Set 5 Set 6 Set 7 Set 8 Set 9
Mean 48.3275 49.1856 48.3719 48.66 47.4775 47.7338 47.5613 47.6231 48.8475Standard Error 0.50234 0.89032 0.70222 0.78343 0.81784 0.94005 0.9694 0.72796 0.7371Median 48.105 48.895 48.49 49.665 46.43 47.83 48.51 47.08 48.08Mode #N/A #N/A #N/A #N/A #N/A #N/A #N/A 46.3 #N/A
Standard Deviation 2.00934 3.56129 2.80888 3.13372 3.27134 3.76022 3.8776 2.91185 2.94839Sample Variance 4.03746 12.68281 7.88979 9.820227 10.70167 14.13923 15.0358 8.47885 8.69298Kurtosis -0.41421 -0.246065 0.618507 -0.12713 -1.596104 -0.488753 -1.064871 0.47499 -0.341887Skewness 0.261972 0.747306 0.283518 -0.836719 0.26528 -0.147822 -0.232884 0.662715 0.517647Range 7.59 12.34 11.65 10.52 8.87 13.38 12.53 11.08 10.28Minimum 44.74 44.17 42.93 42.46 43.16 40.27 41.45 42.68 44.34Maximum 52.33 56.51 54.58 52.98 52.03 53.65 53.98 53.76 54.62Sum 773.24 786.97 773.95 778.56 759.64 763.74 760.98 761.97 781.56
Count 16 16 16 16 16 16 16 16 16
L Berkley DavisCopyright 2009
Central Limit Theorem
48
3
75.016/3
Mean of 16 Sample Means
Mean 48.1987Standard Error 0.20844Median 48.3275
Mode #N/A
Standard Deviation 0.62533Sample Variance 0.391035
Kurtosis -1.480405
Skewness 0.272586Range 1.708125Minimum 47.4775Maximum 49.18563Sum 433.7881
Count 9
Means48.327549.185648.371948.66
47.477547.733847.561347.623148.8475
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
24
Example 7.4: Sampling
3-44
Text Example 3-48
L Berkley DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
25
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 DavisCopyright 2009
MER301: Engineering ReliabilityLecture 7
26
Summary
Functions of Random Variables Linear Combinations of Random Variables Non-Independent Random Variables Non-linear Functions of Random Variables
Central Limit Theorem