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. LetNibe the number of times faceiappears amongthese dice. Describe the joint distribution ofN1andN2.2. A random walk starting at 3 moves up 1 at each step with probability 2/3 and down 1 withprobability 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 or10?4. LetTrbe the number of tosses until therth H in independent coin tosses withpthe probabilityof H on each toss. Write down the probability generating function ofTr.5. LetUrbe the number of tosses until the pattern of ”HTHT....HT” of length 2rappears(meaning the ”HT” is repeatedrtimes in a row). Give a formula forE(Ur).B. Consider a Markov chain (Xn) with transition matrixP. Letfbe a function with numericalvalues defined on the state space of the Markov chain.
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Probability theory, Markov chain, independent coin tosses