We are at a city named Freezo, close to the North Pole, where the days are either "warm" or "cold" (with temperature either above of below freezing). Assume that if we have a cold day today, then tomorrow will be also be cold with probability p (and warm with probability 1-p), and if today is warm then tomorrow will be warm with probability q (and cold with probability 1 - q). Here p and q are some numbers in the interval (0, 1). Consider a matrix 1-q A =D (8a) Prove that X= 1 is an eigenvalue of A (for any choice of p and q). (8b) Set p 3/4 and q= 1/2. Find the eigenvector %3D of A corresponding to X = 1, such that r +y= 1. (8c) It turns out that the numbers x and y that you found in (8b) represent the long-term probability that any given day will be cold and warm, respectively. We are planning a 30-day event Snowtopia" in 2031 that requires at least 18 cold days during the event. Is Freezo a suitable place to host this event? Explain why.
We are at a city named Freezo, close to the North Pole, where the days are either "warm" or "cold" (with temperature either above of below freezing). Assume that if we have a cold day today, then tomorrow will be also be cold with probability p (and warm with probability 1-p), and if today is warm then tomorrow will be warm with probability q (and cold with probability 1 - q). Here p and q are some numbers in the interval (0, 1). Consider a matrix 1-q A =D (8a) Prove that X= 1 is an eigenvalue of A (for any choice of p and q). (8b) Set p 3/4 and q= 1/2. Find the eigenvector %3D of A corresponding to X = 1, such that r +y= 1. (8c) It turns out that the numbers x and y that you found in (8b) represent the long-term probability that any given day will be cold and warm, respectively. We are planning a 30-day event Snowtopia" in 2031 that requires at least 18 cold days during the event. Is Freezo a suitable place to host this event? Explain why.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:We are at a city named Freezo, close to the North Pole, where the days are either "warm" or
"cold" (with temperature either above of below freezing). Assume that if we have a cold day today,
then tomorrow will be also be cold with probability p (and warm with probability 1- p), and if
today is warm then tomorrow will be warm with probability q (and cold with probability 1- q).
Here p and q are some numbers in the interval (0, 1).
Consider a matrix
1-9
A =
1-P
(8a)
Prove that X =1is an eigenvalue of A (for any choice of p and q).
(8b)
Set p = 3/4 and q= 1/2. Find the eigenvector
of A corresponding to A= 1, such that r +y =1
(8c)
It turns out that the numbers x and y that you found in (8b) represent the long-term
probability that any given day will be cold and warm, respectively. We are planning a 30-day event
Snowtopia" in 2031 that requires at least 18 cold days during the event. Is Freezo a suitable
place to host this event? Explain why.
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