PS04_sol - University of Illinois Spring 2010 ECE 313:...

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University of Illinois Spring 2010 ECE 313: Problem Set 4: Solutions Counting Random Variables, Maximum-Likelihood Estimation 1. [The Binomial Random Variable I] (a) p fs = 1 - (1 - 10 - 3 ) 5 (b) p X ( k ) = ( ± 100 k ) p fs k (1 - p fs ) 100 - k , 0 k 100 , 0 , otherwise. (c) 100 × 5 × p fs 2. [Geometric Random Variables] (a) p Y ( k ) = p (1 - p ) k for k 0. (b) E[ Z ] = 1 p = ( 1 . 0526 A student 6 . 6667 C student . (c) 5 = 1 p , so p = 0 . 2 > 0 . 15. Thus the C student might risk not getting a job! (d) P(A does not get an interview in 5 trials)= k =5 p A (1 - p A ) k = (1 - p A ) 5 k 0 =0 p (1 - p ) k 0 = (1 - p A ) 5 = 3 . 125 × 10 - 7 . P(C gets an interview in 5 trials)=1 - (1 - p C ) 5 = 0 . 5563 3. [The Binomial Random Variable II] (a) p X ( k ) = ( ± 8 k ) p k (1 - p ) 8 - k , 0 k 8 , 0 , otherwise. (b) P (undetected error) = p ue = 4 k =1 ± 8 2 k ) p 2 · k (1 - p ) 8 - 2 k 2 . 7832 × 10 - 5 (c) p Y ( k ) = ( ± N k ) p k ue (1 - p ue ) N - k , 0 k N, 0 , otherwise. (d) E[ Y n ] = p ue · n . E[ Y 1 ] 2 . 7832 × 10 - 5 , E[ Y 10 ] 2 . 7832 × 10 - 4 , E[ Y 100 ] 2 . 7832 × 10 - 3 , E[ Y 1000 ] 2 . 7832 × 10 - 2 , E[ Y 10000 ] 2 . 7832 × 10 - 1 4. [Poisson Random Variables] (a) E [ X n ] = λ = p ue · n . E [ X 1 ] 2 . 7832 × 10 - 5 , E [ X 10 ] 2 . 7832 × 10 - 4 , E [ X 100 ] 2 . 7832 × 10 - 3 , E [ X 1000 ] 2 . 7832 × 10 - 2 , E [ X 10000 ] 2 . 7832 × 10 - 1 (b) binomial: p n = 1 - (1 - p ue ) n , p 1 2 . 7832
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This note was uploaded on 04/20/2010 for the course ECE ECE 313 taught by Professor S during the Spring '10 term at University of Illinois at Urbana–Champaign.

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PS04_sol - University of Illinois Spring 2010 ECE 313:...

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