stats150-5 - we see the rst kurrawong. 4 c) Let U be the...

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STAT 150 MIDTERM - SPRING 2007 Instructor: Steven Evans Instructions: Each question part is worth an equal amount. Explain carefully and completely what you are doing at each step in full sen- tences. Quote any results that you use from class. NAME: SID: 1)a) 1)b) 1)c) 2)a) 2)b) 2)c) TOTAL 1
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2 1) Two species of birds, kookaburras and kurrawongs, are sitting along a barbed-wire fence somewhere in Australia. We travel east along the fence from a certain fixed point that we call position 0 and write L ( t ) for the number of kookaburras we have seen after traveling t miles. Similarly, let M ( t ) be the number of kurrawongs we have seen after traveling t miles. Suppose that L and M are independent homogeneous Poisson processes with intensities λ and μ . a) For n = 1 , 2 ,... find the distribution of the distance (in miles) that we have to travel along the fence from position 0 until we see the n th bird.
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3 b) Find the distribution of the number of kookaburras we see before
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Unformatted text preview: we see the rst kurrawong. 4 c) Let U be the distance (in miles) from 0 to the rst bird we see and V be the distance (in miles) from 0 to the rst bird of the other species (that is, if the rst bird we see is a kookaburra, then V is the distance to the rst kurrawong and vice versa ). Find the distribution of V-U . 5 2) Consider the discrete time Markov chain X ,X 1 ,X 2 ,... with state-space { , 1 , 2 , 3 , 4 } and transition matrix 1 / 2 1 / 2 1 / 3 2 / 3 1 / 5 1 / 5 1 / 5 1 / 5 1 / 5 2 / 3 1 / 3 1 a) Which states are persistent and which states are transient? 6 b) What is the approximate value of P { X n = 0 | X = 2 } when n is large? 7 c) Let T = min { n 1 : X n = 1 } . What is E [ T | X = 1]?...
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This note was uploaded on 07/11/2009 for the course CEE cee taught by Professor Monteiro during the Fall '05 term at University of California, Berkeley.

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stats150-5 - we see the rst kurrawong. 4 c) Let U be the...

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