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6 questions related to probability theory and random walks. The questions involve calculating expectations, probabilities, and showing inequalities. The questions are designed for a final exam for a course with course code Math 523A at the University of Washington. The exam is due on June 9th.
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Answer exactly 4 of the following questions:
St+1 =
{ (^) St + 1 p St − 1 1 − p. Let τa = min{t : St = a}, and let a, b be two positive integers. (a) Calculate P(τ−a < τb). (b) Compute E[min{τ−a, τb}].
τc,α = min{t : |St| ≥ ctα^ + 1}. (a) For which c, α is τc,α < ∞ almost surely? (b) For which c, α is Eτc,α < ∞ almost surely?