Stat333_Assign3_09W

# Stat333_Assign3_09W - Stat 333 Winter 2009 Assignment#3 The...

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Unformatted text preview: Stat 333 Winter 2009 Assignment #3 The assignment is due Friday, April 3 , in class. Questions 1–4 posted March 14: These should be done before the term test 1. (a) Letters are drawn randomly with replacement from the set { E, H, R, S, T } . (i) Find the expected number of draws to obtain the sequence “T E S T” (ii) Find the expected numbers of draws required to obtain the sequence “T H R E E T H R E E T H R E E” (b) A coin has probability p of turning up Heads. Find the expected number of tosses required to obtain the sequence “H T H T H H T H T” 2. Consider a Markov chain with state space S = { , 1 , 2 , 3 , 4 } and transition matrix P = 3 4 1 4 1 1 3 1 3 1 3 2 3 1 3 2 5 3 5 (a) Analyze this chain; determine the closed classes and open classes, the recurrent states and transient states, find the equilibrium distributions and periods of the closed classes. (b) Find the absorption probability from each transient state into each closed class. 3. Consider a sequence of repeated independent tosses of a fair coin, each toss resulting in H or T. For each n = 1 , 2 , 3 , . . . define X n = length of the run after the n th toss where a run is a maximal sequence of like outcomes (i.e., all H or all T). For example, if the sequence of outcomes looks like H H T H H H H T . . . then X 1 = 1, X 2 = 2, X 3 = 1, X 4 = 1, X 5 = 2, X 6 = 3, X 7 = 4, X 8 = 1, etc. (a) Model this as a Markov chain by writing down the state space S and the transition matrix P ....
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• Spring '08
• Chisholm
• Probability theory, Exponential distribution, Markov chain, equilibrium distribution, T H R E E T H R E E T H R E E

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Stat333_Assign3_09W - Stat 333 Winter 2009 Assignment#3 The...

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