# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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?
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 $T_{a,b}=\inf\{n>0:s_n=a\hspace{3mm} ... 2answers 619 views ### Stopping Time, Random Walk I'm trying to solve this problem and don't know where to start. If someone could prove it or tell me how or point me to any relevant information I'd very much appreciate it. Let$(s_n)_{n\geq0}$be a ... 1answer 30 views ### Infinite number of 1D-random walkers Place exactly one random walker at each integer in$\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 ... 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$E(\min\{T_a,T_{-b}\})=ab$. ... 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 ...
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 $Y_n$, ...