Let (S, n2 0} be a simple random walk with So = 0 and S, X₁ +...+ X, for n ≥ 1, where 1) = p, P(X;= -1) = q = 1-p for i,i=1,2,... are independent random variables with P(X, 21. Assume pq. Put Fo= {0,0}, F₁ = o(X₁, X2, Xn), n 2 1. Let b, a be two fixed positive tegers. Define T= min{n: S,= -a or S₁=b}. ****
Let (S, n2 0} be a simple random walk with So = 0 and S, X₁ +...+ X, for n ≥ 1, where 1) = p, P(X;= -1) = q = 1-p for i,i=1,2,... are independent random variables with P(X, 21. Assume pq. Put Fo= {0,0}, F₁ = o(X₁, X2, Xn), n 2 1. Let b, a be two fixed positive tegers. Define T= min{n: S,= -a or S₁=b}. ****
A First Course in Probability (10th Edition)
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ISBN:9780134753119
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Chapter1: Combinatorial Analysis
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![5. Let {S, n ≥ 0} be a simple random walk with So = 0 and S₁ = X₁ +...+X for n ≥ 1, where
X₁, i = 1,2,... are independent random variables with P(X; = 1) = p, P(X; = -1) = q = 1 -p for
i > 1. Assume p + q. Put Fo= {2,0}, Fn = 0(X₁, X2, Xn), n ≥ 1. Let b, a be two fixed positive
integers. Define
T= min{n: S₁ = -a or S₁₁=b}.
Determine the probabilities P(ST = -a) and P(ST = b).
We can use the conclusion that:
B)
****
T is a stopping time with respect to the o-fields Fn, n ≥ 0.
A)
Define Zn = (q/p) Sn, n ≥ 0.
{Zn, n ≥ 0} is a martingale with respect to the o-fields
Fn, n ≥ 0.
Assume P(T<∞) = 1.
C)
one can use Doob's Optional Stopping Theorem to conclude
that E[ZT] = 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd14927de-c618-4747-bb5c-8414c571fc63%2F91743be9-ac4a-4e28-b443-d7e611433771%2Frcihmt_processed.png&w=3840&q=75)
Transcribed Image Text:5. Let {S, n ≥ 0} be a simple random walk with So = 0 and S₁ = X₁ +...+X for n ≥ 1, where
X₁, i = 1,2,... are independent random variables with P(X; = 1) = p, P(X; = -1) = q = 1 -p for
i > 1. Assume p + q. Put Fo= {2,0}, Fn = 0(X₁, X2, Xn), n ≥ 1. Let b, a be two fixed positive
integers. Define
T= min{n: S₁ = -a or S₁₁=b}.
Determine the probabilities P(ST = -a) and P(ST = b).
We can use the conclusion that:
B)
****
T is a stopping time with respect to the o-fields Fn, n ≥ 0.
A)
Define Zn = (q/p) Sn, n ≥ 0.
{Zn, n ≥ 0} is a martingale with respect to the o-fields
Fn, n ≥ 0.
Assume P(T<∞) = 1.
C)
one can use Doob's Optional Stopping Theorem to conclude
that E[ZT] = 1.
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