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# HW11sol - University of Illinois Spring 2010 ECE 313...

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University of Illinois Spring 2010 ECE 313: Problem Set 11: Solutions Hazard Rate; Decision Making; Joint Distributions of Random Variables 1. (a) { } { } { } 1 2 Y T X T X T > = > ∪ > . By independence, { } { } { } { } { } 1 2 1 2 PY T P X T P X T P X T P X T > = > + > − > > . (b) Because { } T i P X T e −λ > = , { } ( ) 2 2 2 T T T T T PY T e e e e e −λ −λ −λ −λ − λ > = + = i. [ ] { } ( ) 2 0 0 3 2 2 T T EY PY T dT e e dT −λ − λ = > = = λ ii. Let T p e −λ = . Then, { } 2 2 1/ 2 PY T p p > = − = which has only one solution, 1 1/ 2 p = − in the range 0 ≤ p ≤ 1. Thus, the median value of Y is the solution T to 1 1/ 2 T p e −λ = = − , that is, ( ) 1 ln 2 2 T + . iii. ( ) { } ( ) 2 1 T T Y d f T PY T e e dT −λ −λ =− > = λ for T > 0. Therefore, ( ) ( ) { } ( ) ( ) 2 2 1 1 2 2 2 T T T Y T T T e e e f T hT PY T e e e −λ −λ −λ −λ − λ −λ λ = = = λ > which goes to zero when T goes to zero. (c) The dual-component system has a longer MTBF ( 1 1.5 λ compared to 1 λ ), longer half- life ( ) 1 ln 2 2 λ + compared to ( ) 1 ln 2 λ . and smaller hazard rate than the single component system. 2. (a) (b) ( ) ( ) ( ) 10 1 5 5 10 2 5 u u u o f u e u e f u e Λ = = = Note that ( ) 1 u Λ > for u < 0.2 ln 2. Thus, the likelihood ration test is equivalent to deciding in favor of H 1 if the observed value of X is smaller than the threshold 0.2 ln 2.

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