Homework 4 Solution - EE 351K PROBABILITY & RANDOM...

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Unformatted text preview: EE 351K PROBABILITY & RANDOM PROCESSES FALL 2011 Instructor: Sujay Sanghavi [email protected] Homework 4 Solution Problem 1 There are n multiple-choice questions in an exam, each with 5 choices. The student knows the correct answer to k of them, and for the remaining n- k guesses one of the 5 randomly. Let C be the number of correct answers, and W be the number of wrong answers. (a) What is the PMF of W ? Is W one of the common random variables we have seen in class? (b) What is the PMF of C ? What is its mean, E [ C ] ? Sol : (a) The student guesses one of the 5 choices randomly, probability that a question is guessed correctly = 1 5 . Since the student knows k answers correctly, the number of wrong answers W ∈ [0 ,n- k ] . Therefore P W ( w ) = { ( n- k w ) ( 4 5 ) w ( 1 5 ) n- k- w w ∈ [0 ,n- k ] otherwise which is a binomial random variable. (b) Similarly the PMF of C is P C ( c ) = { ( n- k c- k ) ( 1 5 ) c- k ( 4 5 ) n- c c ∈ [ k,n ] otherwise Then E [ C ] = n ∑ c = k c ( n- k c- k ) ( 1 5 ) c- k ( 4 5 ) n- c = k + n- k 5 . Problem 2 Fischer and Spassky play a sudden-death chess match whereby the first player to win a game wins the match. Each game is won by Fischer with probability p , by Spassky with probability q , and is a draw with probability 1- p- q ....
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This note was uploaded on 02/20/2012 for the course EE 351k taught by Professor Bard during the Spring '07 term at University of Texas at Austin.

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Homework 4 Solution - EE 351K PROBABILITY & RANDOM...

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