Review Markov Chains

Discrete Math Review Markov Chains DO NOT TRY TO DO THE WORK ON THIS SHEET. SHOW YOUR WORK ON SEPARATE PAPER! A (totally bogus and made-up) study found that North Carolina residents tend to move into cities rather than out of them. Suppose that 10% of the Rural population moves into the city each year, while 1% of the Urban population moves out of the city each year. 1.) Draw a weighted digraph representing this situation using Urban (city) and Rural (non-city). 2.) Set up the transition matrix [T] representing this situation. (order = Urban, Rural) 3.) What does the weight on the arrow from Urban to Rural represent? 4.) How long is a transition? 5.) What does T22 represent? 6.) Using [T], exponents, and subscripts, describe HOW you would find the probability that an Urban resident will still be Urban after 7 years. 7.) Suppose your family currently lives in the city. What is the probability they will still live in the city in 7 years? 8.) In the long run, what percentage of North Carolina residents will live in the city? The three largest categories of motor vehicles are Car, Van, and Truck. Every two years, suppose that of the vehicle owners in Winston-Salem who own a Truck, 18% will change to a Van and 20% will change to a Car. Of those who own a Car, 35% will change to a Truck and 20% will change to a Van. Of those who own a Van, 12% will change to a Truck and 32% will change to a Car. 9.) Draw a weighted digraph representing this situation. 10.) Set up the transition matrix [T] representing this situation. (order = Car, Van, Truck) 11.) What does the weight on the arrow from Van to Van represent? 12.) How long is a transition? 13.) What does T23 represent? 14.) Using [T], exponents, and subscripts, describe HOW you would find the probability that your Truck-owning uncle will own a Car in 10 years. 15.) Suppose your uncle currently owns a Truck. What is the probability he will own a Car in 10 years? 16.) In the long run, what percentage of Winston-Salem vehicle owners will own a Van? Two toddlers, Erin and Alonzo, are sharing Cheerios. It is noticed that Erin gives Alonzo a Cheerio 60% of the time and eats it herself the rest of the time. Alonzo, however, only gives Erin a Cheerio 10% of the time and eats it himself the rest of the time. (Assume there is an unlimited number of cheerios. Note: A transition occurs each time a Cheerio is picked up.) 17.) Draw a weighted digraph representing this situation. 18.) Set up the transition matrix [T] representing this situation. (order = Erin, Alonzo) 19.) What does the weight on the arrow from Alonzo to Erin represent? 20.) What does T22 represent? 21.) What does [T]521 represent? 22.) Using [T], exponents, and subscripts, describe HOW you would find the probability that Erin will receive the 3rd Cheerio if Alonzo picked up the 1st one. 23.) Suppose Alonzo got up the 1st Cheerio. What is the probability that Erin will receive the 4th Cheerio? 24.) In the long run, what percentage of the Cheerios will Erin receive? KEY: 1.) 2.) 0.01 Urban U R Rural 0.99 0.90 U R 0.99 0.10 0.01 0.90 0.10 3.) 4.) 5.) 6.) 7.) 8.) 9.) The 1% probability that an Urban resident would move out of the city 1 year The 90% probability that a Rural resident will remain Rural [T]711 [T]711 = 0.94930 = 94.9% probability [T]90 row 1 = 0.9091 = 90.9% will live in an Urban setting 10.) 0.20 0.45 0.32 Car Van 0.56 0.20 0.35 C V T C V T 0.45 0.32 0.20 0.20 0.56 0.18 0.35 0.12 0.62 0.18 0.12 Truck 0.62 11.) 12.) 13.) 14.) 15.) 16.) 17.) The 56% probability that a Van owner will remain a Van owner 2 years The 18% probability that a Truck owner will become a Van owner [T]513 [T]513 = 0.31222 = 31.2% probability [T]50 row 2 = 0.30048 = 30.0% will own a Van 18.) 0.60 Erin Alonzo 0.40 0.90 E A E 0.4 0.6 A 0.1 0.9 0.10 19.) 20.) 21.) 22.) 23.) 24.) The 10% probability that Alonzo will give Erin a Cheerio The 90% probability that Alonzo will keep the Cheerio The 85.5% probability that Alonzo will receive the 5th Cheerio when Erin picks up the 1st one [T]312 [T]312 = 0.13900 = 13.9% probability [T]50 row 1 = 0.14286 = Erin will receive 14.3% of the Cheerios