1
vote
1answer
46 views

{Probability}: choosing keys from a pool without replacement

The OP is trying to understand the following question. The OP understand that if you can always write out the term $$P(X=k) \implies (1-\frac{1}{N})(1-\frac{1}{N-1})\cdots(1-\frac{1}{N-k+1}),$$ ...
0
votes
1answer
56 views

Probability: deviation from the mean

I am having trouble to understand the following. If $S_n=X_1+X_2+......+X_n$, where X_1,X_2 are Bernouli (p). I don't understand this. So you get an intermediate point Constant* sqrt(n). To the ...
2
votes
0answers
57 views

Random walk with $\sum_{n=1}^{\infty} \frac{1}{n} \mathbb{P}\{ S_n > 0 \} < \infty$

Consider a random walk started at $S_0=0$, denoted $S_n = \sum_{k=1}^{n}X_k$, where $X_1$, $X_2$... are the i.i.d increments. If we have $\sum_{n=1}^{\infty} \frac{1}{n} \mathbb{P}\{ S_n > 0 \} ...
0
votes
1answer
60 views

2 dimensional random walk - hit of targets

Consider a random walk in $\mathbb{Z}^2$, $x(j) = x(j-1) + \xi_j$, where the increments are random variables independent and identically distributed with finite support, the expectation $m := ...
0
votes
1answer
63 views

random walk with dependent increment

Consider the following sort of random walk. The position of the walker at time $t$ is represented by the random variable $r(t)$, with $r(0) = 0$. The variable satisfies the following equation, $$ ...
2
votes
0answers
136 views

Finding functions where the increase over a random interval is Poisson distributed

I'm trying to construct a type of function $f(t_1, t_2)$ that counts the number of deterministically simulated Poisson events between two points in time. We can use a single valued function ...
1
vote
0answers
44 views

Random Walk in confined region and loop configurations

Suppose I take a random walk on a 2 dimensional square lattice, but this lattice plane has a finite size, e.g. Dx*Dy. I can not cross the boundary, my step length is the lattice cell size, I either go ...
1
vote
0answers
107 views

Bayesian random walk

Suppose that, at first, I am trying to estimate the mean and standard deviation of some data that I assume to be normally distributed. My prior is gaussian with mean $\mu_0$ and variance $\sigma^2_0$. ...
2
votes
1answer
78 views

law of large number modified statement

The weak law of large number states that, given $Y_n = \sum_{k=1}^{n} X_k$, where $X_k$ are random variables independent and identically distributed with finite expectation $\mu$, $$ \forall ...
0
votes
1answer
85 views

When is an infinite sequence of integers purely deterministic with no randomness involved?

I see in literature very different descriptions of what is a deterministic system such as: "... a system in which no randomness is involved in the development of future states of the system...>>>" I ...
0
votes
2answers
262 views

The variance of a simple random walk/process

I've been trying to wrap my head around this for the past day. Please help! Let $\epsilon_i = \pm 1$ with equal probabilities independently for $i=1,...,N$. Then $Z_i = \epsilon_1 + ... + \epsilon_i$ ...
1
vote
1answer
35 views

Off-lattice Brownian bridges in R^3

Start at a point $(0,0,z_0)$ and take $n$ steps of unit length in a random direction (for each step) in $\mathbb{R}^3$. Let such a walk be valid if the position of the last step, and only the last ...
0
votes
2answers
244 views

How to check that a sequence of numbers is random? [duplicate]

I have a sequence of numbers like 1,7,22,45,12,96,21,45,65,36,85,14,51,16,18,17,16....65... IS there any formula to check whether the sequence is random or not ? In my case odd numbers are ...
2
votes
1answer
194 views

Random walk on finite graph

I know that the stationary distribution of a random walk on the graph is given by, (degree of the node)/($2\times$ total number of links in graph). My question is, how do we get this solution?
4
votes
0answers
79 views

Completeness of random walks in multiple dimensions?

I was reading Artificial Intelligence: Modern Approach (Norvig and Russell), and there was a footnote that really caught my attention. I apologize if the problem is more in the domain of CS than ...
1
vote
2answers
181 views

Is this random walk well studied?

Suppose that we have an ergodic finite Markov chain $C$ with a fintie state space $S$, and we have random variables $X_s$ where $s\in S$. Consider the following random walk $S_0=0$ and $S_{i+1}=S_i ...
1
vote
1answer
103 views

The probability of a discrete-time random process ever incurring a certain drop from “peak to bottom”

Background. Let $Y_1,Y_2,\ldots$ be i.i.d. random variables such that $$P(Y_i<-1) = 0,$$ $$P(Y_i<0) > 0,\quad P(Y_i>0)>0,$$ $$E[Y_i] = \mu > 0\qquad \text{($\mu$ is finite)}.$$ Now ...
1
vote
0answers
185 views

Non-uniform random walk

I'm searching a solution for this problem: Given a segment of length $L$, from $0$ to $L$ divided in $N$ subsegments of the same length, a particle, starting from the subsegment in $x_k$ has a ...
2
votes
2answers
263 views

Are probabilities proportional to the distance traveled in a random walk? What if the initial position is a bit biased?

A marker is placed at zero on the number line and a fair coin is flipped. On each flip we move one unit to the right. If it lands on heads, the marker is moved one unit up. If it lands on tails, the ...