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# hw1 - Boston University Department of Electrical and...

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Unformatted text preview: Boston University Department of Electrical and Computer Engineering EC505 STOCHASTIC PROCESSES Problem Set No. 1 Fall 2008 Issued: Wednesday, Sept. 3, 2008 Due: Friday, Sept. 12, 2008 Problem 1.1 A random experiment consists of tossing a die and observing the number of dots on the top face. Let A 1 be the event that 3 comes up, A 2 the event that an even number comes up, and A 3 the event that an odd number comes up. (a) Find P ( A 1 ) , P ( A 1 ∩ A 3 ). (b) Find P ( A 2 ∪ A 3 ), P ( A 2 ∩ A 3 ), P ( A 1 | A 3 ). (c) Are A 2 and A 3 disjoint? (d) Are A 2 and A 3 independent? Problem 1.2 A group of students is taking a multiple-choice test. For a particular question on the test, the fraction of students who know the answer is p . The fraction that will have to guess the answer is (1- p ). If a student knows the answer, then he will certainly answer the question correctly. If a student doesn’t know the answer and must guess, then the probability of answering the question correctly is 1 /n , where n is the number of choices for the given question. (a) Compute the probability P c that a student who answers the question correctly actually knew the answer. (b) Suppose that the professor believes that p = . 85, i.e. that 85% of the students actually know the answer. Further, suppose that he wants to design the multiple choice question such that P c = . 95, i.e. so that correct answers on the question indicate actual knowledge with 95% probability. How many parts n should the problem have? Problem 1.3 A random variable x has probability distribution function P X ( x ) = 1- e- 2 x u ( x ) where u ( · ) is the unit-step function. (a) Calculate the following probabilities: P [ X ≤ 1] , P [ X ≥ 2] , P [ X = 2] . (b) Find p X ( x ), the probability density function for X . (c) Let Y be a random variable obtained from X as follows: Y = , if X < 2 1 , if X ≥ 2 Find p Y ( y ), the probability density function for Y . Problem 1.4 Let X and Y be statistically independent random variables with probability density functions p X ( x ) = 1 2 δ ( x + 1) + 1 2 δ ( x- 1) , p Y ( y ) = 1 √ 2 πσ 2 exp- y 2 2 σ 2 and let Z = X + Y , and W = XY . 1 (a) Find the conditional probability density functions p Z | X ( z | x =- 1) and p Z | X ( z | x = 1)....
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hw1 - Boston University Department of Electrical and...

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