L Berkley Davis Copyright 2009 MER035: Engineering Reliability Lecture 7 1 MER301: Engineering Reliability LECTURE 7: Chapter 3: Functions of Random Variables, and the Central Limit theorem

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 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 Davis Copyright 2009 MER301: Engineering Reliability Lecture 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…

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 7 4 Functions of Random Variables (3-28) (3-27)

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 7 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

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 7 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

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 7 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…

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 7 8 Non-Independent Random Variables (3-29)

L Berkley Davis Copyright 2009 Independent Random Variables (3-29)

L Berkley Davis Copyright 2009 Non-Independent Random Variables (3-29)

L Berkley Davis Copyright 2009 Functions of Non-Independent Random Variables

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 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 Davis Copyright 2009 MER301: Engineering Reliability Lecture 7 13 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 Davis Copyright 2009 MER301: Engineering Reliability Lecture 7 14 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

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 7 15 Example 7.2: Non-Linear Functions 3-32 and

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 7 16 Random Samples,Statistics and Central Limit Theorem

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 6 17

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 7 18 Example 7.3: Throwing Dice..

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 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=(n 1 =n 2 =even) are not independent P=(1/3)

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 7 20 Central Limit Theorem (3-39)

L Berkley Davis Copyright 2009 Central Limit Theorem Expect for a set of sample data

L Berkley Davis Copyright 2009 Central Limit Theorem

L Berkley Davis Copyright 2009 Central Limit Theorem

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 7 24 Example 7.4: Sampling 3-44 Text Example 3-48

L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 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 Davis Copyright 2009 MER301: Engineering Reliability Lecture 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