632-seppalainen-07spex02

632-seppalainen-07spex02 - 632 Introduction to Stochastic...

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Unformatted text preview: 632 Introduction to Stochastic Processes Spring 2007 Midterm Exam 2 Instructions: Show calculations and give concise justifications for full credit. Points add up to 100. In-Class Part 1. Consider a homogeneous rate Poisson point process on the nonnegative real line [0 , ). Let N ( A ) be the number of points in the subset A of [0 , ). (a) (10 pts) Given that there are 3 points in [0 , t ], what is the probability that there are 5 points in [0 , 2 t ]? In other words, find the conditional probability P { N [0 , 2 t ] = 5 | N [0 , t ] = 3 } . (b) (10 pts) Given that there are 7 points in [0 , 3 t ], what is the probability that there are 5 points in [0 , t ]? In other words, find the conditional probability P { N [0 , t ] = 5 | N [0 , 3 t ] = 7 } . Now color each point blue with probability p and red with probability q = 1- p . Colors of distinct points are independent. (c) (10 pts) Let B 3 be the location of the third blue point. Find the expectation E ( B 3 )....
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This note was uploaded on 08/11/2008 for the course MATH 632 taught by Professor Seppalainen during the Spring '07 term at University of Wisconsin.

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632-seppalainen-07spex02 - 632 Introduction to Stochastic...

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