# PS10 - University of Illinois Fall 2009 ECE 313 Problem Set...

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University of Illinois Fall 2009 ECE 313: Problem Set 10 Exponential and Gaussian Random Variables; Poisson Processes Due: Wednesday November 4 at 4 p.m. Reading: Ross, Chapter 5; Powerpoint Lecture Slides, Sets 24-27 Qfunction : Note available on the COMPASS web page Noncredit Exercises: Chapter 5: Problems 7, 11, 12 15-24, 32, 33; Theoretical Exercises 2, 5, 8, 9; Self-Test Problems 8-11 1. [Reliability function of a triple-modular-redundancy (TMR) system] Consider a triple modular redundancy (TMR) system (cf. Lecture 19 of the Powerpoint slides) with a perfect majority gate. The three modules have lifetimes denoted by X 1 , X 2 , X 3 and fail independently of each other, that is, as in Problem 5 of Problem Set 9, the events { X 1 > t 1 } , { X 2 > t 2 } , and { X 3 > t 3 } are independent events for all choices of t 1 , t 2 , and t 3 . The modules are identical and hence we assume that they have identical reliability functions: P { X i > t } = R ( t ) for i = 1 , 2 , 3. Let Y denote the lifetime of the TMR system. (a) Express the event { Y > t } in terms of unions, intersections and complements of the events { X 1 > t } , { X 2 > t } , { X 3 > t } and use this result to express the reliability function R Y ( t ) of the

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