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Department of Electrical Engineering and Computer Sciences
EECS 126: Probability and Random Processes
Discussion 7
1. Recurrence and Transience of Random Walks
Consider the symmetric random walk Sn = X1 + ··· + Xn in dimension d, where we start at the origin, and with uniform probability we jump to an adjacent point on the d-dimensional lattice Zd. That is, Xi iid∼ Uniform{±e1,…,±ed}, where {e1,…,ed} are the unit coordinate vectors in Rd.
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(c) Use part (b) to show that the random walk for d = 2 is recurrent.
(d) (Optional) Show that the random walk for d = 3 is transient.
(e) Use part (d) to show that the random walk for any d > 3 is also transient.
2. Arrival Times of a Poisson Process
Consider a Poisson process (Nt,t ≥ 0) with rate λ = 1. For i ∈ Z>0, let Ti be a random variable which is equal to the time of the i-th arrival.
(a) Find E[T3 | N1 = 2].
(b) Given T3 = s, where s > 0, find the joint distribution of T1 and T2.
(c) Find E[T2 | T3 = s].
3. Illegal U-Turns
Each morning, as you pull out of your driveway, you would like to make a U-turn rather than drive around the block. Unfortunately, U-turns are illegal and police cars drive by according to a Poisson process with rate λ. You decide to make a U-turn once you see that the road has been clear of police cars for τ > 0 units of time. Let N be the number of police cars you see before you make a U-turn.
(a) Find E[N].
(b) Let n be a positive integer ≥ 2. Find the conditional expectation of the time elapsed between police cars n − 1 and n, given that N ≥ n.
(c) Find the expected time that you wait until you make a U-turn.
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