L Berkley Davis Copyright 2009 MER035: Engineering Reliability Lecture 7 1 MER301: Engineering...

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



2

Summary

Functions of Random Variables Linear Combinations of Random Variables Non-Independent Random Variables Non-linear Functions of Random Variables

Central Limit Theorem



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



4

Functions of Random Variables

(3-28)

(3-27)



5

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



6

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



7

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…



8


P3 T318 82516 78314 73610 6188 547

(3-29)


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)


%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


%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)


Functions of Non-Independent Random Variables

),cov(),cov(),cov(

),cov(),cov(),cov(2

434232

413121

XXXXXX

XXXXXX 22222

4321 XXXXY



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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



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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)



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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



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Example 7.2: Non-Linear Functions

wattsRI 32000802022

222

4

RIPRIP

3-32 and 3-33

3-44



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Random Samples,Statistics and Central Limit Theorem



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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



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact



Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


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


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact

919.0)2515.02485.0( XP

0005.0

2508.0



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Example 7.3: Throwing Dice..



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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



20


(3-39)




Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact


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




Actual Defects

LSL USL


LSL USL

Observed Defects


Product variance


22mactualobs

actual

actual

actual

obs observedobs

tmeasuremenm

actualact


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



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



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Example 7.4: Sampling

3-44

Text Example 3-48



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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.



26

Summary

Functions of Random Variables Linear Combinations of Random Variables Non-Independent Random Variables Non-linear Functions of Random Variables


L Berkley Davis Copyright 2009 MER035: Engineering Reliability Lecture 7 1 MER301: Engineering...

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