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Homework 4 Solution

# Homework 4 Solution - EE 351K PROBABILITY RANDOM PROCESSES...

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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 ] 0 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 ] 0 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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