1. For the simple random walk with p ‡ ½, consider T₁ = min(n ≥ 1, S₂ = 1), and note the following law of total probability: P(T₁ = k) = P(T₁ = k|X₁ = 1) p + P(T₁ = k|X₁ = −1)q, where q = 1-p and which is valid for k≥ 1; special attention needs to be made between k = 1 and k > 1, Using the above, show we can write G₁(t) = pt + qt G₂(t) where G₂(t) = GT₂ (t) and T₂ = min(n ≥ 1, Sn = 2). Recalling that G₂(t) = [G₁(t)]², find G₁(t). 2. Following on from question 1, show that 1 Interpret this result. G₁(1) p> 21/12 - 1²/₁−1 p < 1/1
1. For the simple random walk with p ‡ ½, consider T₁ = min(n ≥ 1, S₂ = 1), and note the following law of total probability: P(T₁ = k) = P(T₁ = k|X₁ = 1) p + P(T₁ = k|X₁ = −1)q, where q = 1-p and which is valid for k≥ 1; special attention needs to be made between k = 1 and k > 1, Using the above, show we can write G₁(t) = pt + qt G₂(t) where G₂(t) = GT₂ (t) and T₂ = min(n ≥ 1, Sn = 2). Recalling that G₂(t) = [G₁(t)]², find G₁(t). 2. Following on from question 1, show that 1 Interpret this result. G₁(1) p> 21/12 - 1²/₁−1 p < 1/1
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.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 43CR
Related questions
Question
N2
![1. For the simple random walk with p#, consider T = min(n > 1, Sm =
1), and note the following law of total probability:
%3D
P(T = k) = P(T1 = k|X1 = 1) p+ P(T1 = k|X1 = -1) q,
%3D
where q = 1 - p and which is valid for k > 1; special attention needs
1 and k > 1,
to be made between k
Using the above, show we can write
G1(t) = pt + qt G2(t)
where G2(t) = Gr,(t) and T, = min(n > 1, S, = 2).
%3D
Recalling that G2(t) = [G1(t)]², find G1(t).
2. Following on from question 1, show that
p>
1
G1(1) =
G-1 p<!
1-p
Interpret this result.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fddfc6eda-000c-4658-af90-eb9f28d739c6%2F4570e12e-b08d-4d1d-bb58-a89a2d8b6b0d%2Fyy0gc2j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. For the simple random walk with p#, consider T = min(n > 1, Sm =
1), and note the following law of total probability:
%3D
P(T = k) = P(T1 = k|X1 = 1) p+ P(T1 = k|X1 = -1) q,
%3D
where q = 1 - p and which is valid for k > 1; special attention needs
1 and k > 1,
to be made between k
Using the above, show we can write
G1(t) = pt + qt G2(t)
where G2(t) = Gr,(t) and T, = min(n > 1, S, = 2).
%3D
Recalling that G2(t) = [G1(t)]², find G1(t).
2. Following on from question 1, show that
p>
1
G1(1) =
G-1 p<!
1-p
Interpret this result.
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