N2 1. For the simple random walk with p#, consider T = min(n > 1, Sm =1), and note the following law of total probability:%3DP(T = k) = P(T1 = k|X1 = 1) p+ P(T1 = k|X1 = -1) q,%3Dwhere q = 1 - p and which is valid for k > 1; special attention needs1 and k > 1,to be made between kUsing the above, show we can writeG1(t) = pt + qt G2(t)where G2(t) = Gr,(t) and T, = min(n > 1, S, = 2).%3DRecalling that G2(t) = [G1(t)]², find G1(t).2. Following on from question 1, show thatp>1G1(1) =G-1 p1-pInterpret this result.

We are given the law of total probability for the first hitting time T1:where q=1-p for k 1.For…

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

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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