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Class 06 Joint distributions and reliability

# Class 06 Joint distributions and reliability - Joint...

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T. R. Willemain MAU Spring 06 1 Joint distributions, independence, and applications to reliability of systems (some bits courtesy of Prof. Pat Sullo)

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T. R. Willemain MAU Spring 06 2 Agenda Joint distributions Independence Application: Reliability of systems In-class exercises Microquiz 06
T. R. Willemain MAU Spring 06 3 Dealing with more than one random variable at once We often have to cope with more than one random variable at once. P(Get 911 call and ambulance free to respond) = ?? f(X=2,Y=7) = ?? When multiple random variables covary, analysis can become difficult or impossible. Toughest cases can be handled by Monte Carlo simulation (next class). Simplifying assumption of “independence” can make analysis feasible.

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T. R. Willemain MAU Spring 06 4 Joint pdf Similarly for joint pmf
T. R. Willemain MAU Spring 06 5 Joint distribution: data and model

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T. R. Willemain MAU Spring 06 6 Joint PDFs and PMFs Say X=thickness, Y=diameter (continuous RVs). Joint pdf f(X,Y) might be any function ≥ 0 whose integral (in this case, volume) = 1. In the special case when X and Y are independent RV’s, joint pdf f(X,Y) simplifies to the product of two one-dimensional pdf’s: f(X,Y) = g(X)*h(Y) When have independence, multiple integrals are simply the product of two or more 1-D integrals, which we already know how to deal with.
T. R. Willemain MAU Spring 06 7 Definition of Independence

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T. R. Willemain MAU Spring 06 8 Joint distributions
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Class 06 Joint distributions and reliability - Joint...

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