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rec10 - Massachusetts Institute of Technology Department of...

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Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Recitation 10 October 12, 2010 Question 1. The two parts of this question are about identities for a probabilistic model with sample space Ω, events A and B , and discrete random variable X . Any time conditioning on an event is indicated, the event has positive probability. An identity is true when it holds without any additional restrictions; it is false when there is any counterexample. 1.1. Which one of the following statements is true ? (a) P ( A B ) may be larger than P ( A ). (b) The variance of X may be larger than the variance of 2 X . (c) If A c B c = , then P ( A B ) = 1. (d) If A c B c = , then P ( A B ) = P ( A ) P ( B ). (e) If P ( A ) > 1 / 2 and P ( B ) > 1 / 2, then P ( A B ) = 1. 1.2. Which one of the following statements is true ? (a) If E [ X ] = 0, then P ( X > 0) = P ( X < 0). (b) P ( A ) = P ( A | B ) + P ( A | B c ) (c) P ( B | A ) + P ( B | A c ) = 1 (d) P ( B | A ) + P ( B c | A c ) = 1 (e) P ( B | A ) + P ( B c | A ) = 1 Question 2. Provide clear reasoning ; partial credit is possible Heather and Taylor play a game using independent tosses of an unfair coin. A head comes up
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