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mid.s06 - Stat 150 Midterm J.P Spring 2006 Student name in...

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Stat 150 Midterm, J.P., Spring 2006. Student name in CAPITALS: A. Warmup. Little explanation required. Just apply results from class notes or text. 1. A Poisson( λ ) number of dice are rolled. Let N i be the number of times face i appears among these dice. Describe the joint distribution of N 1 and N 2 . 2. A random walk starting at 3 moves up 1 at each step with probability 2 / 3 and down 1 with probability 1 / 3. What is the probability that the walk reaches 10 before 0? 3. For the same random walk, what is the expected number of steps until reaching either 0 or 10? 4. Let T r be the number of tosses until the r th H in independent coin tosses with p the probability of H on each toss. Write down the probability generating function of T r . 5. Let U r be the number of tosses until the pattern of ”HTHT .... HT” of length 2 r appears (meaning the ”HT” is repeated r times in a row). Give a formula for E ( U r ). B. Consider a Markov chain ( X n ) with transition matrix P . Let f be a function with numerical values defined on the state space of the Markov chain.
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