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# hw4sol - EE 351K Probability and Random Processes...

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EE 351K Probability and Random Processes FALL 2010 Instructor: Prof. Haris Vikalo [email protected] Homework 4 Solutions 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 ] ? Solution: (a) Assuming the student does not intentionally answer any of the questions he knows incorrectly, we have that W indicates the number of “successes” of a Benoulli random variable, so that P ( W = l ) = n - k l 4 5 l 1 5 n - k - l . We have, then, that W is distributed as a Binomial random variable with parameters ( n - k, 4 / 5) . (b) We have that C = n - W W = n - C . It follows, then, that P ( C = l ) = P ( W = n - l ) = n - k n - l 4 5 n - l 1 5 n - k - ( n - l ) = n - k n - l 4 5 n - l 1 5 l - k . Now, E [ C ] = E [ n - W ] = n - E [ W ] . Since W is a binomial random variable, we have that E [ W ] = ( n - k ) 4 5 . Then E [ C ] = n - 4 5 ( n - k ) = 1 5 ( n + 4 k ) . Problem 2 The runner-up in a road race is given a reward that depends on the difference between his time and the winner’s time. He is given 20 dollars for being one minute behind, 10 dollars for being one to two minutes behind, 5 dollars for being 2 to 6 minutes behind, and nothing otherwise. Given that the difference between his time and the winner’s time is uniformly distributed between 0 and 12 minutes, find the mean and variance of the reward of the runner-up.

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