MER301: Engineering Reliability

1. MER301: Engineering Reliability LECTURE 7: Chapter 3: 3.12-3.13 Functions of Random Variables, and the Central Limit theorem MER035: 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 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… MER301: Engineering Reliability Lecture 7

4. Functions of Random Variables (3-27) (3-28) 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 3-42 3-181 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 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… MER301: Engineering Reliability Lecture 7

8. Non-Independent Random Variables (3-29) MER301: Engineering Reliability Lecture 7

9. Independent Random Variables (3-29)

10. Non-Independent Random Variables (3-29)

11. Functions of Non-Independent Random Variables

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 MER301: Engineering Reliability Lecture 7

13. 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 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 MER301: Engineering Reliability Lecture 7

15. Example 7.2: Non-Linear Functions 3-44 3-32 and 3-33 MER301: Engineering Reliability Lecture 7

16. Random Samples,Statistics and Central Limit Theorem MER301: Engineering Reliability Lecture 7

17. MER301: Engineering Reliability Lecture 6

18. Example 7.3: Throwing Dice.. 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 • EventA= (6,6) given B=(n1=n2=even) are not independent P=(1/3) MER301: Engineering Reliability Lecture 7

20. Central Limit Theorem (3-39) MER301: Engineering Reliability Lecture 7

21. Central Limit Theorem Expect for a set of sample data

22. Central Limit Theorem

23. Central Limit Theorem

24. Text Example 3-48 Example 7.4: Sampling 3-44 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. 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 MER301: Engineering Reliability Lecture 7