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

2.9 Independent Random Variables  Definition

2.9 Independent Random Variables  Mutually Independent  Pairwise Independent

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 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?

2.9 Independent Random Variables  Z=X+Y X i X j … X r are mutually independent

2.9 Problem 1

2.9 Independent Random Variables  Z=max{X,Y}  Z=min{X,Y} X and Y are independent

2.9 Problem 5

3.1 Introduction of Continuous RV CDF pdf

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

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

3.1 Problem 2

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

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

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. e.x. The time we must wait for a new baby is independent of how long we have already spent waiting for him/her Y ~ EXP( )

3.2 Problem 1

3.3 The Reliability and Failure Rate

3.3 Problem 1

Thanks for coming! Questions?