Math Problem Statement
Let T0,m be the time at which the simple random walk reaches 0 for the mth occasion. What is the generating function for T0,m. Show that P(Sn ≤ 0 for for all n ∈ {1, . . . , k} | T0,m = k) = 1 2 m. 1
Solution
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Random Walks
Generating Functions
Ballot Theorem
Formulas
Generating function for the time at which a random walk returns to 0: G(s) = ∑ P(T0 = n) s^n
Generating function for T0,m: G_m(s) = (G(s))^m
P(Sn ≤ 0 for all n ∈ {1, ..., k} | T0,m = k) = 1/2^m
Theorems
Ballot Theorem
Random Walk Recurrence Theorem
Suitable Grade Level
Advanced Undergraduate (Grades 11-12) / Early Graduate Level
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