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Math 523A Final Exam, Exams of Probability and Statistics

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.

Typology: Exams

2020/2021

Uploaded on 05/11/2023

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University of Washington Math 523A
Final exam
Due by: June 9
Answer exactly 4 of the following questions:
1. Let Stbe a simple random walk on Zand set τ= min{t:|St| a}for some integer a > 0.
Calculate Eτ2.
Hint : Consider a martingale involving the fourth power of St.
2. Consider a biased random walk on the integers: for some 0 <p<1,
St+1 =St+ 1 p
St1 1 p.
Let τa= min{t:St=a}, and let a, b be two positive integers.
(a) Calculate P(τa< τb).
(b) Compute E[min{τa, τb}].
3. In a sequence of fair coin tosses, let τ011 and τ000 denote the number of tosses until witnessing
the corresponding pattern.
(a) Find the value of Emax{τ011, τ000}.
(b) Find Eτ2
01.
4. A gambler plays the following game. In each round, he can pay any 0 < p < 1 dollars, and
win $1 with probability p(independently). Show that the probability that the gambler’s net
gain exceeds hat any of the first nrounds is at most exp(h2/2n).
5. Consider percolation on a binary tree with parameter p < 1
2. Let Cdenote the cluster of the
root. Show that
P(|C| > k)exp(kα(p))
for some α(p)>0 (find α(p) explicitly).
6. Let Stbe a simple random walk on Z, and for c, α > 0 define
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?

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University of Washington Math 523A

Final exam

Due by: June 9

Answer exactly 4 of the following questions:

  1. Let St be a simple random walk on Z and set τ = min{t : |St| ≥ a} for some integer a > 0. Calculate Eτ 2. Hint: Consider a martingale involving the fourth power of St.
  2. Consider a biased random walk on the integers: for some 0 < p < 1,

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}].

  1. In a sequence of fair coin tosses, let τ 011 and τ 000 denote the number of tosses until witnessing the corresponding pattern. (a) Find the value of E max{τ 011 , τ 000 }. (b) Find Eτ 012.
  2. A gambler plays the following game. In each round, he can pay any 0 < p < 1 dollars, and win $1 with probability p (independently). Show that the probability that the gambler’s net gain exceeds h at any of the first n rounds is at most exp(−h^2 / 2 n).
  3. Consider percolation on a binary tree with parameter p < 12. Let C denote the cluster of the root. Show that P(|C| > k) ≤ exp(−kα(p)) for some α(p) > 0 (find α(p) explicitly).
  4. Let St be a simple random walk on Z, and for c, α > 0 define

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?