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Unformatted text preview: Boston University Department of Electrical and Computer Engineering EC505 STOCHASTIC PROCESSES Problem Set No. 1 Solutions 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? Solution: (a) P ( A 1 ) = 1 / 6 , P ( A 1 ∩ A 3 ) = P ( A 1 ) = 1 / 6. (b) P ( A 2 ∪ A 3 ) = P (Ω) = 1; P ( A 2 ∩ A 3 ) = P ( ∅ ) = 0, P ( A 1 | A 3 ) = P ( A 1 ∩ A 3 ) P ( A 3 ) = 1 / 3 (c) Yes. (d) Clearly not. P ( A 2 ∩ A 3 ) = P ( ∅ ) = 0 6 = P ( A 2 ) P ( A 3 ). 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? Solution: (a) The events K and C correspond to the events of student knowing the answer and anwering it correctly. Thus: P c = P ( K | C ) And by Bayes’ Rule: P c = P ( C | K ) P ( K ) P ( C ) = P ( C | K ) P ( K ) P ( C | K ) P ( K ) + P ( C | ¯ K ) P ( ¯ K ) Subsitute P ( C | K ) = 1, P ( C | ¯ K ) = 1 n , P ( K ) = p , and P ( ¯ K ) = 1- p P c = p 1- p n + p 1 (b) Solve for n in the above equation. n = P c (1- p ) p (1- P c ) Then subsitute P c = . 95 and p = . 85 n = . 95( . 15) . 85( . 05) = 3 . 353 But it is best to have a question with an integer number of choices, so round n up to 4. 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 ....
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- Fall '09
- Probability theory, probability density function, pdf estimate