Probability and Statistics in Aerospace Engineering

NASA / TP--1998-207194

Probability and Statistics in Aerospace Engineering

M.H. Rheinfurth and L.W. Howell Marshall Space Flight Center, Marshall Space Flight Center, Alabama

National Aeronautics and Space Administration Marshall Space Flight Center

March 1998

NASA Center for AeroSpace Information 800 Elkridge Landing Road Linthicum Heights, MD 21090-2934 (301) 621-0390

Available from: ii

National

Technical Information Service 5285 Port Royal Road Springfield, VA 22161 (703) 487-4650

TABLE OF CONTENTS

I.

INTRODUCTION

.....................................................................................................................

1

A. Preliminary Remarks ........................................................................................................

1

B. Statistical Potpourri ..........................................................................................................

1

C. Measurement Scales .........................................................................................................

2

D. Probability and Set Theory ...............................................................................................

2

II.

PROBABILITY .........................................................................................................................

5

A. Definitions of Probability .................................................................................................

5

B. Combinatorial Analysis (Counting Techniques) ...............................................................

6

C. Basic Laws of Probability ................................................................................................

10

D. Probability Distributions ..................................................................................................

19

E. Distribution (Population) Parameters ...............................................................................

23

F. Chebyshev's Theorem ......................................................................................................

26

G. Special Discrete Probability Functions .............................................................................

27

H. Special Continuous Distributions .....................................................................................

32

I. Joint Distribution Functions .............................................................................................

41

J. Mathematical Expectation ................................................................................................

48

K. Functions of Random Variables ........................................................................................

50

L. Central Limit Theorem (Normal Convergence Theorem) ................................................

61

M. Simulation (Monte Carlo Methods) ..................................................................................

61

IH.

STATISTICS ..............................................................................................................................

64

A. Estimation Theory ............................................................................................................

64

B. Point Estimation ...............................................................................................................

65

C. Sampling Distributions .....................................................................................................

74

D. Interval Estimation ...........................................................................................................

79

E. Tolerance Limits ...............................................................................................................

83

E Hypothesis/Significance

Testing ......................................................................................

85

G. Curve Fitting, Regression, and Correlation ......................................................................

91

H. Goodness-of-Fit Tests .......................................................................................................

103

I. Quality Control .................................................................................................................

107

J. Reliability and Life Testing ..............................................................................................

112

K. Error Propagation Law .....................................................................................................

118

BIBLIOGRAPHY

.................................................................................................................................

124

iii

LIST OF FIGURES

, Venn diagram .............................................................................................................................

11

2. Conditional probability ..............................................................................................................

11

3. Partitioned sample space ...........................................................................................................

15

4. Bayes' Rule ................................................................................................................................

15

5. Cartesian product .......................................................................................................................

19

o Function A _ B ..........................................................................................................................

20

7. Coin-tossing experiment ............................................................................................................

21

8. Probability function diagram .....................................................................................................

21

9. Cumulative distribution function ...............................................................................................

22

10. Location of mean, median, and mode ........................................................................................

24

11. Chebyshev's theorem .................................................................................................................

26

12. Normal distribution areas: two-sided tolerance limits ...............................................................

33

13. Normal distribution areas: one-sided tolerance limits ...............................................................

34

14. Uniform p.d.f .............................................................................................................................

37

15. Examples of standardized beta distribution ...............................................................................

39

16. Gamma distribution ...................................................................................................................

39

17. Cantilever beam .........................................................................................................................

42

18. Posterior distribution with no failures .......................................................................................

46

19. Two tests and one failure ...........................................................................................................

46

20. Lower confidence limit ..............................................................................................................

47

21. A function of a random variable ................................................................................................

51

22. Random sine wave .....................................................................................................................

52

23. Probability density of random sine wave ..................................................................................

53

24. Probability integral transformation ............................................................................................

54

25. Sum of two random variables ....................................................................................................

57

26. Difference of two random variables ..........................................................................................

58

27. Interference random variable .....................................................................................................

59

28. Buffon's needle ..........................................................................................................................

62

29. Area ratio of Buffon's needle ....................................................................................................

62

30. Sampling distribution of biased and unbiased estimator ...........................................................

67

31. Estimator bias as a function of parameter .................................................................................

69

V

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