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Questions tagged [random-walk]

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

A random walk is a type of stochastic process with random increments, and it is usually indexed by a continuous time variable or an equally spaced discrete time variable.

An elementary example of a random walk is the random walk on $\mathbb{N}_0$, which starts at $0$ and at each step moves $+1$ or $−1$ with equal probability. The path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler can all be approximated by random walk models, even though they may not be truly random in reality.

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1D Random walk with state-dependent move probability

We start with $a + b$ balls in the urn: $a$ white and $b$ black ones.At each step we randomly choose a ball and if it's white then $m$ of the black balls in the urn turn white, symetrically for choosing black one. We put the chosen ball back in the…
micsza
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Random walk with floor at $0$

This weekend my SO and I were trying to analyze a problem which boiled down to a 1-dimensional random walk with a floor at 0. That is, starting at 0, move $\{-1, +1\}$ with probability $\frac 1 2$ each, but anything which would go to -1 instead…
Bobson
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Simple random walk on a finite network

We have a simple random walk S on a finite set of $\mathbb{N}: {0,...,N}$. Let $m_x$ be the expected number of steps, starting at $x$, required to reach $0$ or $N$ for the first time. Show that $m_x$ satisfies the conditions: (a) $m_0 = 0$ (b) $m_N…
Vydai
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Probability of one outcome in random walk

This question is really throwing me off: Lets say there's two players, A and B. Each game consists of betting \$1. Gameplay ends when one player has all of the money. Player A starts with \$3, B starts with $5. If Player A has probability of winning…
Grant Seltzer
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Random walk and expected value

Consider a classical symmetric random walk that starts at origin $x=0$ and lasts for $N$ steps. If $x=-1$ is reached, a walk is terminated. What is the expected value of this process? I divided a walk into 2 parts. A path hits $x=-1$ at some step…
qoqosz
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Expected value of a random walk given value at previous time

I have a homework question dealing with random walks. One part of the question required me to find the probabilities $p$ and $q$ such that $$E[M_1] = M_0$$ which I achieved. Having found these, the next part requires me to use these values to find…
Newtt
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Random Walk Limit Behavior

Suppose $\{ X_t \}$ is a sequence of i.i.d. random variables, with support $\{-1,1\}$ and distribution $P(1)=P(-1)=1/2$. Thus, $S_t = \sum_{s=1}^{t} X_s$ is a zero mean random walk. Also, $S_t$ is a martingale, but the conditions for Doob's…
Jong
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What is the measure for a Random Walk?

Let $F$ be a distribution on $\mathbb{Z}$. Let $(X_1,X_2,...)$ be an i.i.d. sequence of random variables with distribution $F$. Then $S_0=0, S_1=X_1, S_2=X_1+X_2,...$ is called the random walk with step size distribution $F$. When I then have an…
mathfemi
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Random walk and powers of 2

I'm in trouble on the following problem: given a random walk starting at point N on the integer number line, how many steps should I wait before the walk hits a power of two at least once, with probability $P$ with $P\gt 0$?
Riccardo.Alestra
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