The immortal jellyfish Turritopsis dohrnii is a sea creature that spends the majority of its life in two states: the immature form called a "polyp" which is attached to the seafloor and the mature form called a "medusa" which swims freely in the water. Polyps develop into free-swimming medusa as they age, and medusae (the plural form of medusa) are able to return to their polyp state when they are stressed. On a given day, a polyp develops into a medusa with a probability of and a medusa becomes a polyp again with a probability of 3. Pk mk (a) For the state * = with the first element the probability of being a polyp and the second the probability of being a medusa, draw a state transition diagram and write the transition matrix A for this process. (b) Compute the probability that an individual starting as a polyp is a medusa after two days. (c) Find a diagonal matrix D and an invertible matrix T so that A = TDT-¹. (d) Find the coefficients c₁, c₂ needed to write the initial probability vector po = B combination po = c₁v₁ + C₂v₂ where v₁, 02 are eigenvectors of A. (e) Find the fixed probability vector p*= limk+ Pk for this Markov chain. as the linear
The immortal jellyfish Turritopsis dohrnii is a sea creature that spends the majority of its life in two states: the immature form called a "polyp" which is attached to the seafloor and the mature form called a "medusa" which swims freely in the water. Polyps develop into free-swimming medusa as they age, and medusae (the plural form of medusa) are able to return to their polyp state when they are stressed. On a given day, a polyp develops into a medusa with a probability of and a medusa becomes a polyp again with a probability of 3. Pk mk (a) For the state * = with the first element the probability of being a polyp and the second the probability of being a medusa, draw a state transition diagram and write the transition matrix A for this process. (b) Compute the probability that an individual starting as a polyp is a medusa after two days. (c) Find a diagonal matrix D and an invertible matrix T so that A = TDT-¹. (d) Find the coefficients c₁, c₂ needed to write the initial probability vector po = B combination po = c₁v₁ + C₂v₂ where v₁, 02 are eigenvectors of A. (e) Find the fixed probability vector p*= limk+ Pk for this Markov chain. as the linear
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:The immortal jellyfish Turritopsis dohrnii is a sea creature that spends the majority of its life in two
states: the immature form called a "polyp" which is attached to the seafloor and the mature form called
a "medusa" which swims freely in the water. Polyps develop into free-swimming medusa as they age,
and medusae (the plural form of medusa) are able to return to their polyp state when they are stressed.
On a given day, a polyp develops into a medusa with a probability of and a medusa becomes a polyp
again with a probability of 3.
Pk
mk
(a) For the state ™k = with the first element the probability of being a polyp and the second the
probability of being a medusa, draw a state transition diagram and write the transition matrix A
for this process.
(b) Compute the probability that an individual starting as a polyp is a medusa after two days.
(c) Find a diagonal matrix D and an invertible matrix T so that A = TDT-¹.
(d) Find the coefficients C₁, C₂ needed to write the initial probability vector Po
combination po = c₁v₁ + C₂02 where V₁, V2 are eigenvectors of A.
(e) Find the fixed probability vector p*= limk→ Pk for this Markov chain.
=
B
as the linear
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