0
votes
0answers
36 views

Probability of Stopping Time Taking specific value - Random Walk 1d

We are considering a simple random walk $(X_n)_{n\in\mathbb{N}}$ starting at $X_0=0$ with $X_n=\sum_{i=1}^nY_i$ where $Y_i$ are iid and $\mathbb{P}(Y_i=1)=\mathbb{P}(Y_i=-1)=\frac{1}{2}$. We want to ...
2
votes
1answer
54 views

Random walk around circle

For one of the exercises of my homework I need to answer the following question, but I am not sure how I should apply gamblers ruin theory to solve this problem (it is stated as a hint, not that I ...
0
votes
1answer
49 views

Probability of random walk traversal

Consider a random walk on an connected, non-bipartite, undirected graph G. Show that, in the long run, the walk will traverse each edge with equal probability. Note: The walk can traverse each edge ...
0
votes
1answer
21 views

Find a asymptotic upper bound for $\sum_{n=N}^{\infty}p_{ii}^{(n)}$ for a asymetric one-dimensional simple random walk

For asymmetric one-dimensional simple random walk, that is $$P(X_n = X_{n-1} + 1) = p = 1 - P(X_n = X_{n-1} - 1)$$ for some $p \ne 1/2$, provide an asymptotic upper bound for ...
1
vote
0answers
159 views

Random walk - expected distance not from origin

We have an assignment on random walk, but I can't figure out the expected value. The situation is as follows: In the origin there is a hunter that shoots at a duck, but misses. The duck starts at a ...
0
votes
1answer
76 views

Strong Law of Large numbers, prove expression is Standard Normal

Question: "Let $X_{1},X_{2},\cdots$ be a sequence of independent random variables such that $X_{n}$ is binomial with parameters $2n-1$ and $p=\frac{1}{2}$. If $$Y_{n} = \frac{2(X_{1}+X_{2}+\cdots ...
3
votes
3answers
86 views

A fly on a triangle?

A fly is on the vertex of a triangle. It can move left with probability $\frac 12$ and right with probability $\frac 12$. What is the expected number of moves till it reaches its starting point? ...
0
votes
1answer
197 views

Simple Probability Matrix

Consider a simple model that predicts whether you pass you next test or not based on the result of your previous test. If you pass your previous test, then you have 0.2 chance you will pass your ...
1
vote
2answers
363 views

Probability related to random walks in two dimensions

I'm trying to show that two random walks will eventually meet in a two dimensional setting but I can't figure out where to start. Can someone lead me towards the right direction?
0
votes
2answers
147 views

Random Walk Proof Problem

I have to do the following problem: Let $(s_n)_{n\geq 0 }$ be a 1-dimensional, unbiased random walk. For $a,b\in\mathbb Z$, let $T_a=\inf\{n>0:s_n=a\}$ and $T_{a,b}=\inf\{n>0:s_n=a\hspace{3mm} ...
1
vote
2answers
619 views

Stopping Time, Random Walk

I'm trying to solve this problem and don't know where to start. If someone could prove it or tell me how or point me to any relevant information I'd very much appreciate it. Let $(s_n)_{n\geq0}$ be a ...
3
votes
1answer
30 views

Infinite number of 1D-random walkers

Place exactly one random walker at each integer in $\Bbb Z$ and define $Y_n$ as the number of these who are at the origin at time n. Show that $0<\displaystyle\lim_{n\to\infty}P\{Y_n=0\}<1$ and ...
2
votes
1answer
126 views

A problem about symmetric random walk

Consider a symmetric random walk $P(X_i=1)=P(X_i=-1)=1/2$, $S_0=0$, $T_a=\min(n:S_n=a)$ We already know that $P(T_a>T_{-b})=1-P(T_{-b}< T_a)=\frac{b}{a+b}$ and $E(\min\{T_a,T_{-b}\})=ab$. ...
3
votes
2answers
306 views

Expected value of function of random walk

I am trying to calculate $\lim_{n \to \infty} {E[e^{i \theta \frac{S_n}{n}}]}$. Where $\theta \in \mathbb{R}$, and $S_n$ is simple random walk. I could simplify it to $\lim_{n \to \infty}E[\cos(\theta ...
0
votes
1answer
330 views

Probability mass function of a random walk process

Let $Y_n$ be a random walk process defined as $Y_n = Y_{n-1} + X_n$; $n = 1,2\ldots$ and $Y_0 = 0$, where $X_k = +1$ with probability $p$ and $-1$ with probability $1-p$. Write down the pmf for $Y_n$, ...