# Probability and Stochastic Processes: Problems

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Connexions module: m11261 1 Probability and Stochastic Processes: Problems ∗ Don Johnson This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License † Exercise 1 Joe is an astronaut for project Pluto. The mission success or failure depends only on the behavior of three major systems. Joe feels that the following assumptions are valid and apply to the performance of the entire mission: • The mission is a failure only if two or more major systems fail. • System I, the Gronk system, fails with probability 0.1. • System II, the Frab system, fails with probability 0.5 if at least one other system fails. If no other system fails, the probability the Frab system fails is 0.1. • System III, the beer cooler (obviously, the most important), fails with probability 0.5 if the Gronk system fails. Otherwise the beer cooler cannot fail. 1 What is the probability that the mission succeeds but that the beer cooler fails? 2 What is the probability that all three systems fail? 3 Given that more than one system failed, determine the probability that: 3.1 The Gronk did not fail. 3.2 The beer cooler failed. ∗ Version 1.3: May 24, 2004 10:45 pm -0500 † http://creativecommons.org/licenses/by/1.0 http://cnx.org/content/m11261/1.3/ Connexions module: m11261 2 3.3 Both the Gronk and the Frab failed. 4 About the time Joe was due back on Earth, you overhear a radio broadcast about Joe while watching the Muppet Show. You are not positive what the radio announcer said, but you decide that it is twice as likely that you heard "mission a success" as opposed to "mission a failure". What is the probability that the Gronk failed? Exercise 2 The random variables X1 and X2 have the joint pdf ∀b1,b2, b1 ∧ b2 > 0 : ( p X1,X2 (x1, x2 ) = 1 pi2 b1b2( b1 2 + x12 ) ( b2 2 + x22 )) (1) 1 Show that X1 and X2 are statistically independent random variables with Cauchy density functions. 2 Show that ΦX1

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