Math Problem Statement
Suppose a simple random walk is at height a at time n ≥ 1. What is the generating function for the random time before the process reaches height a − 1.
Solution
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Math Problem Analysis
Mathematical Concepts
Random Walk
Generating Functions
Recurrence Relations
Boundary Conditions
Formulas
G_a(z) = \mathbb{E}[z^{T_{a-1}}]
G_a(z) = \frac{1}{2} z G_{a+1}(z) + \frac{1}{2} z G_{a-1}(z)
G_a(z) = \left( \frac{1 - \sqrt{1 - z^2}}{z} \right)^a
Theorems
Theory of Simple Random Walks
Generating Function Theorem
Symmetry in Random Walks
Suitable Grade Level
Undergraduate Level
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