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es RR, RW, and WW, re w.

es RR, RW, and WW, re w.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Next Generation Red Pink White A given plant species has red, pink, or white flowers according to the genotypes RR, RW, and WW, respectively. If each type of these genotypes is crossed with a pink-flowering plant (genotype RW), then the transition matrix is as shown to the right. Red 0.5 0.5 This Pink 0.25 0.5 0.25 Generation White 0 0.5 0.5 Assuming that the plants of each generation are crossed only with pink plants to produce the next generation, show that regardless of the makeup of the first generation, the genotype composition will eventually stabilize at 25% red, 50% pink, and 25% white. (Find the stationary matrix)
Transcribed Image Text:Next Generation Red Pink White A given plant species has red, pink, or white flowers according to the genotypes RR, RW, and WW, respectively. If each type of these genotypes is crossed with a pink-flowering plant (genotype RW), then the transition matrix is as shown to the right. Red 0.5 0.5 This Pink 0.25 0.5 0.25 Generation White 0 0.5 0.5 Assuming that the plants of each generation are crossed only with pink plants to produce the next generation, show that regardless of the makeup of the first generation, the genotype composition will eventually stabilize at 25% red, 50% pink, and 25% white. (Find the stationary matrix)
Label the transition matrix P. Since P = |, which has only entries, this Markov chain is Markov chain. (Type an integer or decimal for each matrix element.) Explain what is important about the stationary matrix/matrices because this Markov chain belongs to this special category of Markov chains. Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. O A. It must have a unique stationary matrix, which in this case is (Type an integer or decimal for each matrix element.) O B. It must have a series of stationary matrices, characterized +k( ), wherek is a real number from 0 to 1. O C. It cannot have a stationary matrix. Explain how to conclude that, regardless of the first generation, the genotype composition will eventually stabilize at 25% red, 50% pink, and 25% white. Choose the correct answer below. O A. Another important property of this type of Markov chain is that given any initial-state matrix So, the state matrices S, approach the stationary matrix. O B. Another important property of this type of Markov chain is that random chance will eventually produce a generation that perfectly matches the stationary state, at which point every future generation will identically match the stationary state. O C. Another important property of this type of Markov chain is that given any initial-state matrix So, the state matrices S always return to So after a finite number of steps.
Transcribed Image Text:Label the transition matrix P. Since P = |, which has only entries, this Markov chain is Markov chain. (Type an integer or decimal for each matrix element.) Explain what is important about the stationary matrix/matrices because this Markov chain belongs to this special category of Markov chains. Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. O A. It must have a unique stationary matrix, which in this case is (Type an integer or decimal for each matrix element.) O B. It must have a series of stationary matrices, characterized +k( ), wherek is a real number from 0 to 1. O C. It cannot have a stationary matrix. Explain how to conclude that, regardless of the first generation, the genotype composition will eventually stabilize at 25% red, 50% pink, and 25% white. Choose the correct answer below. O A. Another important property of this type of Markov chain is that given any initial-state matrix So, the state matrices S, approach the stationary matrix. O B. Another important property of this type of Markov chain is that random chance will eventually produce a generation that perfectly matches the stationary state, at which point every future generation will identically match the stationary state. O C. Another important property of this type of Markov chain is that given any initial-state matrix So, the state matrices S always return to So after a finite number of steps.
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