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(a) Derive the difference equation for this problem. (b) Derive the partial differential equation governing this process if k→ 0 and h→0 such that lim kho h (c) Assume that the probability of staying in place is zero and the initial position of the random walker is known with certainty. Determine the most likely location of the random walker and the associated probability at the following three time steps: i. N₁ = 1 x 105 ii. N₂ = 5 x 105 iii. N₂ = 1 × 106 (d) Using your numerical intuition, what is the most likely location of the random walker after an arbitrary number of time steps.

(a) Derive the difference equation for this problem. (b) Derive the partial differential equation governing this process if k→ 0 and h→0 such that lim kho h (c) Assume that the probability of staying in place is zero and the initial position of the random walker is known with certainty. Determine the most likely location of the random walker and the associated probability at the following three time steps: i. N₁ = 1 x 105 ii. N₂ = 5 x 105 iii. N₂ = 1 × 106 (d) Using your numerical intuition, what is the most likely location of the random walker after an arbitrary number of time steps.

Advanced Engineering Mathematics
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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Similar to the previous problem, suppose that in a one-dimensional random walk, at each time step h the probability of moving to the right k is a, and the probability of moving to the left k is also a. The probability of staying in place is b. (a) Derive the difference equation for this problem. (b) Derive the partial differential equation governing this process if k→ 0 and h→0 such that k2 lim kh 10 h K. (c) Assume that the probability of staying in place is zero and the initial position of the random walker is known with certainty. Determine the most likely location of the random walker and the associated probability at the following three time steps: i. N₁ = 1 x 105 ii. N₂ = 5 x 105 iii. N₂ = 1 x 106 (d) Using your numerical intuition, what is the most likely location of the random walker after an arbitrary number of time steps.
Transcribed Image Text:Similar to the previous problem, suppose that in a one-dimensional random walk, at each time step h the probability of moving to the right k is a, and the probability of moving to the left k is also a. The probability of staying in place is b. (a) Derive the difference equation for this problem. (b) Derive the partial differential equation governing this process if k→ 0 and h→0 such that k2 lim kh 10 h K. (c) Assume that the probability of staying in place is zero and the initial position of the random walker is known with certainty. Determine the most likely location of the random walker and the associated probability at the following three time steps: i. N₁ = 1 x 105 ii. N₂ = 5 x 105 iii. N₂ = 1 x 106 (d) Using your numerical intuition, what is the most likely location of the random walker after an arbitrary number of time steps.
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