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Conica, Cui Yuanyuan

Tutorial 4 Cover: C2.9 Independent RV C3.1 Introduction of Continuous RV C3.2 The Exponential Distribution C3.3 The Reliability and Failure Rate Assignment 4. Conica, Cui Yuanyuan. Definition. 2.9 Independent Random Variables. 2.9 Independent Random Variables. Mutually Independent

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Conica, Cui Yuanyuan

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  1. Tutorial 4Cover:C2.9 Independent RV C3.1 Introduction of Continuous RV C3.2 The Exponential Distribution C3.3 The Reliability and Failure Rate Assignment 4 Conica, Cui Yuanyuan

  2. Definition 2.9 Independent Random Variables

  3. 2.9 Independent Random Variables • Mutually Independent • Pairwise Independent

  4. 2.9 Independent Random Variables • Pairwise independent of a given set of random events does not imply that these events are mutual independence . • Example Suppose a box contains 4 tickets labeled by 112 121 211 222 Let us choose 1 ticket at random, and consider the random events A1={1 occurs at the first place} A2={1 occurs at the second place} A3={1 occurs at the third place} P(A1)=? P(A2)=? P(A3)=? P(A1A2)=? P(A1A3)=? P(A2A3)=? P(A1A2A3)=? QUESTION: By definition, A1,A2, and A3 are mutually or pairwise independent?

  5. 2.9 Independent Random Variables • Z=X+Y Xi Xj …Xr are mutually independent

  6. 2.9 Problem 1

  7. 2.9 Independent Random Variables • Z=max{X,Y} • Z=min{X,Y} X and Yare independent

  8. 2.9 Problem 5

  9. 3.1 Introduction of Continuous RV CDF pdf

  10. F(x) 1 x 1 f(x) 1 x 1 3.1 Introduction of Continuous RV Example

  11. F(x) f(x) 1 x 1 1 x 1 3.1 Introduction of Continuous RV

  12. 3.1 Problem 2

  13. 3.2 The Exponential Distribution X ~ EXP() CDF & pdf

  14. 3.2 The Exponential Distribution X ~ EXP() Example • Interarrival time; • Service time; • Lifetime of a component; • Time required to repair a component.

  15. 3.2 The Exponential Distribution X ~ EXP() Memoryless Property X – Lifetime of a component t – working time until now Y – remaining life time The distribution of Y does not depend on t. Y ~ EXP() e.x. The time we must wait for a new baby is independent of how long we have already spent waiting for him/her

  16. 3.2 Problem 1

  17. 3.3 The Reliability and Failure Rate

  18. 3.3 Problem 1

  19. Thanks for coming!Questions?

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