# Tagged Questions

For questions on random walks, a mathematical formalization of a path that consists of a succession of random steps.

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### probability of completing a self-avoiding chessboard tour

Someone asked a question about self-avoiding random walks, and it made me think of the following: Consider a piece that starts at a corner of an ordinary $8 \times 8$ chessboard. At each turn, it ...
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### simplify summation of factorial (random walk)

I suspect that the expression $$\sum_{n=0}^N \frac{(N-2n)^2}{n!(N-n)!}$$ simplifies to $$\frac{2^N}{(N-1)!}$$ But I cannot find the intermediate steps. Can someone give me a hint how I can deduce ...
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### Expected number of times Random Walk crosses 0 line.

Suppose we have a simple random walk: $$x_t = x_{t-1} + \epsilon_{t}$$ Where $$\epsilon_{t} = iid\ \mathcal{N} (0,1)$$ Assume that x starts at ...
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### Mean distance from origin after $N$ equal steps of Random-Walk in a $d$-dimensional space.

I am looking for a formula that evaluates the mean distance from origin after $N$ equal steps of Random-Walk in a $d$-dimensional space. Such a formula was given by "Henry" to a question by "Diego" (q/...
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### Expected travel of random walk in arbitrary game with multiple payouts

As explained here, the average distance or 'travel' of a random walk with $N$ coin tosses approaches: $$\sqrt{\dfrac{2N}{\pi}}$$ What a beautiful result - who would've thought $\pi$ was involved! ...
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### First player to win k matches

A series of matches are held between n identical competitors. Each is won by one of the n with equal probability (no ties). I'm looking for a probabilistic description of the outcome when looking at ...
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### How long until everyone has been in the lead?

Earlier, I asked a question about a series of competitions: A series of matches are held between n identical competitors. Each is won by one of the n with equal probability (no ties). I'm looking ...
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### Recurrence of a certain class of $2$-$d$ random walks

As is well known, a symmetric random walk on $\mathbb{Z}^d$ (the lattice of $d$ dimensional vectors with integer components) is recurrent if and only if $d=1,2$. In particular it is transient for $d=3$...
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### If I wanted to randomly find someone in an amusement park, would I be faster roaming around or standing still?

Assumptions: The other person is constantly and randomly roaming Foot traffic concentration is the same at all points of the park Field of vision is always the same and unobstructed Same walking ...
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### 1D random walk-probability to go back to origin

Suppose There is a random walk starting in origin while probability to move right is 1/3 and probability to move left 2/3.What is the probability to return to the origin. Thank you
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### Expected number of steps in a random walk with a boundary

Let's say I am trying to climb a flight of $N$ stairs. Each time I want to take a step, I flip a fair coin. Heads means I take a step up; tails means I take a step down. If I'm at the bottom of the ...
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### Probability of a biased random walk hitting an absorbing barrier in some number of steps

Let's say I have a biased random walk over the integers in some interval [0, L] where the endpoints of the interval ('0' and 'L', respectively) are fully absorbing. The walker starts at some position ...
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### How can we directly see that the number of random walks starting and ending at the origin is ${n\choose n/2}^2$?

In an infinite two-dimensional square-shaped grid, we define four directions, north, south, east, west. We thus have $4^n$ random walks of length $n$. If we end where we started, for every north step ...
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### What's the easiest way to show that a random walk can go arbitrarily far?

Let's consider the simplest situation. On the one dimensional line of integers, and we starts from the origin. Each time we either move left or right (at the same probability) for 1 unit. How do I ...
I got stuck today trying to understand an argument of the Frank den Hollander Book's. The problem is described below. Let $S_n=\sum_{i=1}^n X_i$ be the simple random walk in $\mathbb{Z}^d$, that is ...