Compute Random Walk Hitting and Return Probabilities
Problem: Random Walk Probabilities
Consider simple symmetric random walks on integer lattices.
Part 1: One-Dimensional Absorbing Boundaries
A particle starts at an integer position x with absorbing boundaries at integers a and b, where a < x < b. At each step, the particle moves one unit to the left or right, each with a probability of 1/2.
What is the expected number of steps until the particle first hits either boundary a or boundary b?
Part 2: Two-Dimensional Return Probability
A particle starts at the origin (0, 0) on the integer lattice. At each step, it moves one unit up, down, left, or right, each with a probability of 1/4.
What is the probability that the particle eventually returns to the origin?
Part 3: Three-Dimensional Return Probability
Extend the previous question to three dimensions: the particle starts at the origin (0, 0, 0) and at each step moves to one of its six neighboring lattice points, each with a probability of 1/6.
What is the probability that the particle eventually returns to the origin?
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Compute Random Walk Hitting and Return Probabilities
Problem: Random Walk Probabilities
Consider simple symmetric random walks on integer lattices.
Part 1: One-Dimensional Absorbing Boundaries
A particle starts at an integer position x with absorbing boundaries at integers a and b, where a < x < b. At each step, the particle moves one unit to the left or right, each with a probability of 1/2.
What is the expected number of steps until the particle first hits either boundary a or boundary b?
Part 2: Two-Dimensional Return Probability
A particle starts at the origin (0, 0) on the integer lattice. At each step, it moves one unit up, down, left, or right, each with a probability of 1/4.
What is the probability that the particle eventually returns to the origin?
Part 3: Three-Dimensional Return Probability
Extend the previous question to three dimensions: the particle starts at the origin (0, 0, 0) and at each step moves to one of its six neighboring lattice points, each with a probability of 1/6.
What is the probability that the particle eventually returns to the origin?