3. An office worker owns 3 umbrellas which she uses to go from home to work and vice versa. If it is raining in the morning she takes an umbrella (if she has one) on her trip to work and if it is raining at night she takes an umbrella (if she has one) on her trip home. If it is not raining, she doesn't take an umbrella. At the beginning of each trip it is raining with probability p = 1 − q, independently of previous trips. Let X represent the number of umbrellas available at the beginning of the n-th trip and assume that 0 < p < 1. (a) Explain why {Xn}n≥1 is a Markov Chain with state space S having the transition probability matrix P: = {0, 1, 2, 3} P = ( 0 0 0 1 0 0 1-p P 0 1-p Р 0 P Р 0 0 (b) Classify the states of the Markov chain in (a) under additional condition 0 ≤ p≤1. (c) Explain or calculate directly why P(X3 = 0 | X₁ = 0) = 1 − p. (d) If no umbrellas are available at the beginning of the first trip, calculate the average number of trips until there are again no umbrellas available. (e) Explain why the limits of P(Xn calculate them. = i) as n∞ exists, for i = 0, 1, 2, 3, and (f) After a large number of trips, how many umbrellas are available on average at the beginning of a trip? (g) In a large number of trips, about what proportion of journeys does she get wet?
3. An office worker owns 3 umbrellas which she uses to go from home to work and vice versa. If it is raining in the morning she takes an umbrella (if she has one) on her trip to work and if it is raining at night she takes an umbrella (if she has one) on her trip home. If it is not raining, she doesn't take an umbrella. At the beginning of each trip it is raining with probability p = 1 − q, independently of previous trips. Let X represent the number of umbrellas available at the beginning of the n-th trip and assume that 0 < p < 1. (a) Explain why {Xn}n≥1 is a Markov Chain with state space S having the transition probability matrix P: = {0, 1, 2, 3} P = ( 0 0 0 1 0 0 1-p P 0 1-p Р 0 P Р 0 0 (b) Classify the states of the Markov chain in (a) under additional condition 0 ≤ p≤1. (c) Explain or calculate directly why P(X3 = 0 | X₁ = 0) = 1 − p. (d) If no umbrellas are available at the beginning of the first trip, calculate the average number of trips until there are again no umbrellas available. (e) Explain why the limits of P(Xn calculate them. = i) as n∞ exists, for i = 0, 1, 2, 3, and (f) After a large number of trips, how many umbrellas are available on average at the beginning of a trip? (g) In a large number of trips, about what proportion of journeys does she get wet?
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter12: Probability
Section12.3: Conditional Probability; Independent Events; Bayes' Theorem
Problem 10E
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And please do not copy other's work,very appreciate!!
And please do not copy other's work,very appreciate!!
Transcribed Image Text:3. An office worker owns 3 umbrellas which she uses to go from home to work and vice
versa. If it is raining in the morning she takes an umbrella (if she has one) on her
trip to work and if it is raining at night she takes an umbrella (if she has one) on
her trip home. If it is not raining, she doesn't take an umbrella. At the beginning
of each trip it is raining with probability p = 1 − q, independently of previous trips.
Let X represent the number of umbrellas available at the beginning of the n-th
trip and assume that 0 < p < 1.
(a) Explain why {Xn}n≥1 is a Markov Chain with state space S
having the transition probability matrix P:
=
{0, 1, 2, 3}
P =
(
0
0
0 1
0
0 1-p P
0 1-p
Р 0
P Р
0
0
(b) Classify the states of the Markov chain in (a) under additional condition 0 ≤
p≤1.
(c) Explain or calculate directly why P(X3 = 0 | X₁ = 0) = 1 − p.
(d) If no umbrellas are available at the beginning of the first trip, calculate the
average number of trips until there are again no umbrellas available.
(e) Explain why the limits of P(Xn
calculate them.
=
i) as n∞ exists, for i = 0, 1, 2, 3, and
(f) After a large number of trips, how many umbrellas are available on average at
the beginning of a trip?
(g) In a large number of trips, about what proportion of journeys does she get
wet?
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