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M118 Weeks 14 &amp; 15

# M118 Weeks 14 &amp; 15 - NOW I SNOT THE TI ME TO S TART...

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NOW IS NOT THE TIME TO START SLACKING!!! CHAPTER 8 IS DIFFICULT AND IS HEAVILY WEIGHTED ON THE FINAL EXAM! YOUR WEBWORK ASSIGNMENTS FROM NOW UNTIL THE END OF THE SEMESTER WILL HAVE REVIEW PROBLEMS FROM CHAPTERS 1 THROUGH 3 !!

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CHAPTER 8 - MARKOV CHAINS CHAPTER 8 - MARKOV CHAINS TWO CLASSIC EXAMPLES: A.) An experiment consists of watching DJ White shoot free-throws, each timenoting whether or not he hits the shot. If he makes the current shot, he has a 70% chance of making the next shot, and if he misses the current shot, he has a 40% chance of hitting the next shot. Q: If hemakes thecurrent shot, what is theprobability that he also makes the shot after the next one?
B.) An experiment consists of watching a rat in a maze with three compartments, noting every 15 minutes which compartment the rat is in. If the Rat is in compartment A on the current observation, then it is equally likely to bein each of the three compartments on the next observation. If the rat is in compartment B on the current observation, then it is equally like to stay there as to moveto another compartment, and if it moves, it is equally likely to be in each of the other two compartments. Finally, if therat is in compartment C on thecurrent observation, it is always in compartment A on the next observation. Q: If the rat is in compartment A now, what is the probability that it is in compartment C 30 minutes from now?

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Q: WHAT MAKES THESE TWO EXPERIMENTS MARKOV CHAINS? DEFN: An experiment is a Markov Chain Markov Chain if it satisfies: a.) At each stageof the experiment, theoutcome is one of a fixed number of states (state = possible outcome at each stage), and b.) The conditional probability of moving from one state to another on the next observation (observation = stageof experiment) depends only on the two states in question and nothing else. NOTE: Theexperiment usually STARTS in one of the possible states.
TERMINOLOGY: 1.) Wenumber the states 1, 2, up to number of states, and 2.) Pr[moving to state k | in state j now ] = p jk = transition probability transition probability FREE-THROW EXAMPLE RAT IN MAZE EXAMPLE state1 = miss shot state1 = in compartment A state2 = hit shot state2 = in compartment B state3 = in compartment C p 11 = .60 p 12 = .40 p 11 = 1/3 p 12 = 1/3 p 13 = 1/3 p 21 = .30 p 22 = .70 p 21 = 1/4 p 22 = 1/2 p 23 = 1/4 p 31 = 1.0 p 32 = 0 p 33 = 0

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TWO METHODS OF SUMMARING THE INFO IN A MARKOV CHAIN: A.) Transition Diagram Transition Diagram - a diagram that shows each of the possible states along with the probabilities of moving between states on the next observation. B.) Transition Matrix - a matrix that has as entries the conditional probabilities of moving from the state in that row of the matrix into the state that is in that column of the matrix on thenext observation. NOTE: Entries in the transition matrix are the p jk ’s from before!
EG) FIND THE TRANSITION DIAGRAM AND MATRIX FOR THE TWO EXAMPLES DONE THUS FAR: FREE-THROW EXAMPLE RAT IN MAZE EXAMPLE state1 = miss shot state1 = in compartment A state2 = hit shot state2 = in compartment B state3 = in compartment C p 11 = .60 p 12 = .40 p 11 = 1/3 p 12 = 1/3 p 13 = 1/3 p 21 = .30 p 22 = .70 p 21 = 1/4 p 22 = 1/2 p 23 = 1/4 p 31 = 1.0 p 32 = 0 p 33 = 0

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M118 Weeks 14 &amp; 15 - NOW I SNOT THE TI ME TO S TART...

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