0
votes
0answers
47 views

Stationary distribution for random walks on directed graph

There is an equation (Eq. (2)) in reference by Lovasz and Winkler about the stationary distribution of a random walk on directed graphs that I would like to find references for where the equation is ...
2
votes
2answers
38 views

a probability question related to computing the variance of a specific pattern

With respect to a given sequence of points $\{X_1, ... X_t, ...X_n\}$. I can understand why $E[S]= \frac{n-1}{2}$. But how to get that $Var[S]$.
2
votes
1answer
27 views

Ant problem with discrete combinatorical background.

an ant can move along a grid in $\mathbb{Z}^2$. But the ant can only go upwards and to the right(with equal probability). The ant starts in the point $(0,0)$, but there is an electrical wire from ...
10
votes
1answer
195 views

Ruin time for a two-input “risk only” slot machine

Imagine a "risk only" slot machine that takes 'coins' corresponding to some real number fraction of a dollar $p$, returns the coin with probability $p$, and eats the coin with probability $(1-p)$. ...
1
vote
2answers
223 views

Probability of visiting state $s_1$ of a Markov chain more than $N$ times in $L$ steps.

Assume we have a two-state Markov chain, with $s_1$ and $s_0$ denoting the two states. The initial state of the Markov chain is either $s_1$ or $s_0$ with probability $p_1$ or $p_0$, respectively. The ...
2
votes
2answers
135 views

Probability of a Specific Run Occurring in a Random Process

A random process has three possible outcomes: $A$, $B$, and $C$. At each step, the outcome is decided randomly, and is uncorrelated with previous outcomes. The outcomes occur with probabilities $p_A$, ...
1
vote
2answers
50 views

Finding the probability of a client getting the same token in two consecutive interactions.

I am trying to find the probability in the following real-world inspired scenario. If I have a finite set of whole numbers from 0 to 4 billion which I call tokens and $n$ clients. Each time a client ...
6
votes
2answers
529 views

A question on calculating probabilities for the random walk

I am currently working on a high school project revolving around the 'Cliff Hanger Problem' taken from ”Fifty Challenging Problems in Probability with Solutions” by Frederick Mosteller. The problem ...
1
vote
1answer
101 views

Generating function

We define $Z_i=\max\{X_i,X_i'\}$ where $X_i$ and $X_i'$ are i.i.d. random variables. We would like to know the generating function of $Z_i$ in terms of the generating function of $X_i$, which is ...
8
votes
1answer
372 views

Throwing balls into $b$ buckets: when does some bucket overflow size $s$?

Suppose you throw balls one-by-one into $b$ buckets, uniformly at random. At what time does the size of some (any) bucket exceed size $s$? That is, consider the following random process. At each of ...
0
votes
2answers
621 views

Lottery “Sum” forecasting

I was wondering if anyone can provide some mathematical insights to forecasting the "SUM" in this link as a time series. It is an oscillatory, range bound and poisson distribution. How can Monte Carlo ...
4
votes
1answer
318 views

Counting process which is not a Poisson process

Please construct a counting process N, whose r.v. N(t) are distributed as Poisson(λt) but the process N itself is not a Poisson process. This is an assignment in our Stochastic Process class. So I ...
2
votes
2answers
312 views

Collisions in random walk in $\mathbb{Z}^n$

Given a set $S$ of $r$ points in $\mathbb{Z}^n$, $S=(p_1,p_2,p_3.., p_r)$ , each a starting point for random walk with step size 1. What is the probability they will all eventually meet at the same ...