What is Random walk: Definition and 96 Discussions
In mathematics, a random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.
An elementary example of a random walk is the random walk on the integer number line,
Z
{\displaystyle \mathbb {Z} }
, which starts at 0 and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler: all can be approximated by random walk models, even though they may not be truly random in reality.
As illustrated by those examples, random walks have applications to engineering and many scientific fields including ecology, psychology, computer science, physics, chemistry, biology, economics, and sociology. Random walks explain the observed behaviors of many processes in these fields, and thus serve as a fundamental model for the recorded stochastic activity. As a more mathematical application, the value of π can be approximated by the use of a random walk in an agent-based modeling environment. The term random walk was first introduced by Karl Pearson in 1905.Various types of random walks are of interest, which can differ in several ways. The term itself most often refers to a special category of Markov chains, but many time-dependent processes are referred to as random walks, with a modifier indicating their specific properties. Random walks (Markov or not) can also take place on a variety of spaces: commonly studied ones include graphs, others on the integers or the real line, in the plane or higher-dimensional vector spaces, on curved surfaces or higher-dimensional Riemannian manifolds, and also on groups finite, finitely generated or Lie. The time parameter can also be manipulated. In the simplest context the walk is in discrete time, that is a sequence of random variables (Xt) = (X1, X2, ...) indexed by the natural numbers. However, it is also possible to define random walks which take their steps at random times, and in that case, the position Xt has to be defined for all times t ∈ [0,+∞). Specific cases or limits of random walks include the Lévy flight and diffusion models such as Brownian motion.
Random walks are a fundamental topic in discussions of Markov processes. Their mathematical study has been extensive. Several properties, including dispersal distributions, first-passage or hitting times, encounter rates, recurrence or transience, have been introduced to quantify their behavior.
Hello Everyone,
The following is a subproblem of research project I'm working on, i.e. not a homework. Let's suppose you have a bounded 2d plane and n distinct probes that do random-walk in that plane. The world is closed in a sense that a probe going outside the border ends up being on the...
Hi all:
Now I have a question about the concept of diffusion coefficient in two cases: the diffusion equation (J=DdT/dx) and the random walk (tao^2=6Dt). My quesion is the two D in two equations are the same or different. If they are different, is there any relationship between them?
Best
Xu
Hi guys,
I'm doing some thinking about random walk.
Imagine there is a bounded 2D plane and a single spawn point. The spawn produces units which must bring in minerals scattered around the spawn. The locations of minerals are not known, so the units diffuse randomly away from the spawn...
Suppose X is a random walk with probability
P(X_k=+1)=p and P(X_k=-1)=q=1-p
and S_n=X_1+X_2+...+X_n
Can anyone explain why does line 3 equal to line 4?
P(S_k-S_0≠0 ,S_k-S_1≠0 ,…,S_k-S_{k-1}≠0)
=P(X_k+X_{k-1}+⋯+X_1≠0 ,X_k+X_{k-1}+⋯+X_2≠0 ,…,X_k≠0)
=P( X_k≠0 ,X_k+X_{k-1}≠0...
Hi guys,
I was reading about random walks and i encountered one step of a proof which i don't know how to derive in a mathematically rigorous way.
the problem is in the attached file and S is a random walk with X_i as increments, X_i =
{-1,+1}
I know that intuitively we can switch the...
I've been trying to find the solution to the following problem but it's evaded me thus far.
Take the classic one dimension random walk scenario. I start at point 0 and can either step forward +1 step or step backwards -1 step (equal probability). I can countinue like this for N steps.
If...
I'm reading a stat textbook and it says the following:
Let a discrete-time random walk be defined by Xt = Xt-1 + et, where the et's are i.i.d. normal(0,σ2). Then for t≧1,
(i) E(Xt) = 0
(ii) Var(Xt) = t σ2
However, the textbook doesn't have a lot of justifications for these results and...
Homework Statement
Hello, I was reading Feynman's lectures on physics, and I'm having trouble following some deductions in the part about Probability. The random walk is a problem in which someone starts at x = 0 ant then takes a step forward (x = 1) or backward (x = -1) and after N steps de...
Question about "random walk"
Homework Statement .
.. Recall that in a random walk where each step
has length l, the total distance traveled after N steps is L = N1/2lHomework Equations
The Attempt at a Solution
My problem is the N number of steps, not sure how i would find that.
I saw in the...
Hello folks, this is my first post. I'm not quite sure that I fully understand the idea of a random walk of a photon that is generated at the core of a star.
I've read this:
http://www3.wooster.edu/physics/jrIS/Files/Walker_Web_article.pdf
My understanding of the theory is this:
A...
I have created a code to simulate a scaled random walk, but I need to now modify it to generate 1000 paths and then estimate the variance with n = 100, for t = 0.25, 0.5, 0.75 and 1. Can anyone help out with how to generate all 1000 paths? Sorry if this is painfully easy, I am a complete...
Homework Statement
Let us define the following random walk:
Start with a randomly chosen (with uniform probability) unit vector (with respect to the usual Euclidean norm) in \mathbb{R}^n and call it x^{(0)}. Now, x^{(t+1)} is generated from x^{(t)} in the following manner - randomly and...
Hello
I am trying to find an analytic expression for the probability that a particle will have passed a position x after a time t.
It is a 1D random walk, the probability distribution after time t that a particle will end at position x is given by a gaussian, but I need to know how many...
Homework Statement
Consider the following random walk on the integers:
\mathbb{S}=\{0, 1, 2, 3, ... , L\}
Let Wn = {the state k \in\mathbb{S} you are in after the n'th transition}
\mathbb{P}[W_{n+1}= k \pm 1 | W_{n} = k] = \frac{1}_{2}
For \ 1\leq k\leq L-1
Otherwise...
Hi, I'm trying to read into CTRW, but I'm finding the information online a little difficult to take in. From what I've read the process differs from normal random walks in that jumps take place after some waiting time \tau, which can be from 0<\tau<\infty. Would I also be right in saying that...
Hi, I have a problem
(1) where I need to compute the ratio of probabilities of hitting and stopping at a positive vertical barrier x vs hitting and stopping at a negative horizontal barrier y after starting from (0,0).
I feel that by symmetry, the answer to this would be the same as...
Homework Statement
A clown stands at the side of a swimming pool. In his hand is a bag containing n red balls and n blue balls. At each step he puts his hand into the bag and pulls out a random ball and throws it away. If the ball is red, he makes a step towards the pool and if it is blue...
Hi,
I'm modeling receptors moving along a cell surface that interact with proteins inside of a cell. I figured it would be easier to model the receptors in spherical coordinates, however I'm unsure of how to model a random walk. In cartesian coordinates, I basically model a step as:
x = x +...
Hello,
I have read the probability chapter in Feynman's lectures on physics. And got fascinated by the random walk. There is a statement, that in a game where either a vertical distance of +1 or -1 can be walked each move, the expected value of the absolute distance (lets call it <D>) from...
Problem: Three friends do a 1D Random Walk with p=q starting at 0. Are they guaranteed to all meet again somewhere? So all the friends are starting at 0 at the same time and we want to know if they are guaranteed to meet again anywhere in the walk.
Hint: if they do meet, which places are...
I have a problem concerning a one-dimensional random walk in potentials. Assume a one-dimensional space [0,1] and a probability distribution p(x). At every point x we have a probability p(x) to go left and 1-p(x) to go right. Assume some smooth distribution of p(x) with boundaries p(0) = 0 and...
I have the intrinsic noise PSD spec for Honeywell's QA3000 accels, and I need to find the velocity random walk due to noise to show that I'm within my allowable. The PSD is given in units of g^2/Hz vs Hz. I'm given a velocity random walk due to noise spec of 15 (micro-m/s)/rt(s) that I must be...
This is for my Statistics and Stochastic Processes class, we are learning about random walk
[b]1. Let p<q where p is success (+1) and q is failure (-1). Let T=last time we hit 0 (T\geq0) Find P(T=t)
Then using the answer from the above question, make up a formula for \sum(2n choose...
My impression from looking through the internet is, that it is a well-accepted opinion that stock market calculations are based on the random walk hypothesis.
A friend of mine says otherwise. Could you provide some opinions or rather references to prove one view or the other?
Homework Statement
T_{x} is the passage time min{n : x(n) = x} for paths starting from x(0)=0
Show:
\lim_{x\rightarrow +\infty}E_{0}(e^\frac{-\alpha\\T_{x}}{x^2}) = e^\-\sqrt{2\alpha}
for any \alpha ≥ 0
Homework Equations
The Attempt at a Solution
I came up with the...
Hi,
I know that the expectation E(Sn) for a one-dimensional simple random walk is zero. But what about the variance?
I read in http://en.wikipedia.org/wiki/Random_walk#One-dimensional_random_walk" that the variance should be E(Sn2) = n.
Why is that? Can anyone prove it?
Thank you...
Homework Statement
A particle's Compton wavelength arises because of the "smearing out" effect of quantum uncertainty, which can be modeled by a random walk or by Brownian noise. I think I understand this, but what I do not get is how do you arrive at a precise frequency of oscillation from...
Homework Statement
I need to determine the standard deviation from the mean position value <x> in a random walk scenario. We are given that the probability of taking a step to the right is p and the probability of taking a step to left is q. Please note that p and q need not be equal. The...
Hi guys, Would you be so kind to give me a hand with the below, please?
Basically I need a formula that can help me solve the following:
Let's say I have a pond with many many frogs and I have been measuring the amount they can jump.
The frogs jumps seem to follow a normal distribution...
How often do you return AND CROSS the origin in a random walk?
Is there a probability distribution for how many times can you expect a change in the lead of a coin tossing contest between two players (heads Vs. tails)?
Which is the most likely # of changes in the lead?
A random walk can be defined by the following recurrence relation,
X_{t+1} = X_t + \Delta X
where \Delta X \sim \mathcal{N}(0, \sigma^2).
At any time t1 \geq 0 a strategist may enter, and at any time t2 > t1 a strategist may exit. The resulting profit p is given by:
p(t1,t2) = X_{t2}...
I understand the mean square end-to-end distance of a polymer of stepsize b and number of units N is given by
\left\langle R^{2}\right\rangle = \sum_{i,j=1}^{N} \left\langle r_{i} r_{j}\right\rangle + \left\langle \sum_{i \neq j}^{N} r_{i} r_{j}\right\rangle
And that the cross terms drop...
Hi - I am trying to find the probability of hitting one of two boundaries in a simple random walk (I describe the problem precisely below). Actually, my main concern is to find the probability distribution over time to hit either one of two boundaries. I think that this is a very standard...
A grid of 4x4 is given
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A ball is located at the center of the grid which is to perform a 5 step random walk with equal probability in any...
Homework Statement
Say a drunk starts making his steps of equal distance from a lamppost. Assuming that each of the steps are of equal distance, and N as the total number of steps, what is the probability of him/her ending at the lamppost? Find the probability when N is even and also for...
Starting out at zero on a number line and moving in succession one unit right or left at random, what is the probability that you will eventually return to zero?
Homework Statement
http://img171.imageshack.us/img171/4711/52047019pc2.png
Homework Statement
Homework Equations
Random walk equations: Xi is i.i.d. and S_n = X_1+...+X_n and S0=0
The Attempt at a Solution
I don't have a idea where to begin. I couldn't start because I don't...
Homework Statement
Let (X1, X2, ..., Xn,...) be iid increments (with mean µ and variance ∂^2) of a random walk Sn=X1+X2+...+Xn. What are the expected value, variance of Sn?
Prove that lim n-> ∞ Sn =+ ∞ if µ>0 and lim n-> ∞ Sn =- ∞ if µ<0
Homework Equations
The Attempt at a...
Homework Statement
Two drunks start out together at the origin, each having equal probability of making a step to the left or right along the x-axis. Find the probability that they meet again after N steps. It is understood that the men make their steps simultaneously.
Homework Equations...
For the random walk approach to gases, liquids, or solids, why isn't there a gradient? The atoms don't jump by themselves, right? They should have to feel forces to jump...
The question:
"Two men start out together at the origin, each having a 1/2 chance of making a step to the left or right along the x-axis. Find the probability that they meet again after N steps."
It then says it may help to consider their relative position but I don't see how that would...
The question:
"Two men start out together at the origin, each having a 1/2 chance of making a step to the left or right along the x-axis. Find the probability that they meet again after N steps."
It then says it may help to consider their relative position but I don't see how that would...
I was unaware that one could obtain the Dirac equation as a result of a random walk. I believe that this has been done by other researchers, but I found it by Ord's papers:
http://arxiv.org/find/quant-ph/1/au:+Ord_G/0/1/0/all/0/1
Anyone else find these interesting?
My interest is due to...
Ok I have a varaition of the random problem as follows. We have a container with volume V and N particles. We consider a subvolume v and n particles. The probability of particles being inside v is (v/V)
Ok I found the mean of n (mean number of molecules in v)
< n > = N*v*(1/V)
Then they...
We know that the standard deviation [sig] for a random walk, represented by a net distance d, to be approximately the square root of the total number of steps N, each of length L, from the origin. I. e., d~N1/2L~[sig]L.
Does the angle attained after these steps also have a significant...