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Discover the concept of independent random variables, mutual vs. pairwise independence, exponential distribution, reliability, and failure rate. Learn through examples and problems.
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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
Definition 2.9 Independent Random Variables
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 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?
2.9 Independent Random Variables • Z=X+Y Xi Xj …Xr are mutually independent
2.9 Independent Random Variables • Z=max{X,Y} • Z=min{X,Y} X and Yare independent
F(x) 1 x 1 f(x) 1 x 1 3.1 Introduction of Continuous RV Example
F(x) f(x) 1 x 1 1 x 1 3.1 Introduction of Continuous RV
3.2 The Exponential Distribution X ~ EXP() CDF & pdf
3.2 The Exponential Distribution X ~ EXP() Example • Interarrival time; • Service time; • Lifetime of a component; • Time required to repair a component.
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
