Tagged Questions
0
votes
0answers
13 views
Importance sampling of finite path of stochastic difference equation
Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...
0
votes
0answers
15 views
Find the distribution of the increments between two consecutive steps of the walk from a Langevin equation?
Given a Langevin equation of a stochastic process: $X_{I+1}=X_I-F(X_I)+W_I$ - where $F(X_I)$ is a position dependent force, and $W_I$ is the Wiener process term (i.e., Gaussian / white-noise).
How do ...
6
votes
4answers
332 views
Is winning a soccer match independent of previous wins\losses?
$\quad$ I have a friend of mine who is a bit of a gambler ask me this question. He is of poor mathematical background, but has a sense of logic and will probably accept a logical answer in the natural ...
2
votes
1answer
55 views
description of a Wiener Process assuming a Laplace Distribution
Is there a description of the Wiener Process when a Laplace distribution is assumed rather than a normal one?
0
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0answers
38 views
Ergodicity, mixing and stationarity
When considering sequences, I know that mixing and stationarity implies ergodicity. This is a stationary mixing process is ergodic. But, if I have a stationary and non-ergodic sequence, can I conclude ...
2
votes
0answers
56 views
Distribution/expected length of the shortest path in infinite random geometric graphs
Consider an infinite random geometric graph $G(\rho,d)$ in which vertices are uniformly and independently scattered over the 2D plane with density $\rho$ and edges connect the vertices that are closer ...
8
votes
1answer
126 views
Density of robots doing random walk in an infinite random geometric graph
Consider an infinite random geometric graph in which the node locations follow a Poisson point process with density $\rho$ and edges are placed between the nodes that are closer than $d$. Therefore, ...
0
votes
0answers
44 views
Comparing two different leagues of similar but not equal distributions around a standard deviation of error of a prediction from a rating system
This query ties a lot of my interests in rating sports teams together, because as I’ve mentioned before I do a version of the Kenneth Massey method (as per his 1997 thesis ...
4
votes
1answer
159 views
Distribution of arrival times to server for an M/M/1 queue (what the server experiences)
In an M/M/1 queue, we know that inter-arrival times are exponentially distributed, and that service times are the same. What is the distribution of to-server inter-arrival times (aka service start ...
0
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0answers
44 views
About Galton watson process
My question is about a homework question that I found interesting. It gives another proof (without using martingales) for that the critical Galton Watson tree dies out eventually. But it has given a ...
0
votes
0answers
98 views
Building a probability distribution function from observation
There are N players and M objects, each of the objects has a value. Each player has a strategy in choosing an object. Each round a player will choose an object, many players can choose the same ...
1
vote
0answers
88 views
Should the intersection of 2 event sets be on the same sample space?
I'm trying to understand a stochastic process and consecutive events in time.
Should the intersection of two events be on the same sample space, on a combination of simultaneous sample spaces, or a ...
2
votes
0answers
64 views
Modeling a 1D random walk with nonconstant probability in a point
There is a 1D discrete random walk system which the probability of all points are 1/2(probability of going forward and backward) except one point which locate on l (l is an integer number).the ...
1
vote
1answer
167 views
Poker winrate statistical test
In the game of poker, how do we test $H_o : \textrm{winrate} > 0$? Or how can we test $H_o : \textrm{winrate} = x$ for $x \in \mathbb{R}$? Winrate might be measured as the population $ profit per ...
0
votes
1answer
64 views
Error metrics to assess a stochastic model with Monte Carlo simulations
Suppose we have constructed a model of some stochastic system; we are also able to perform Monte Carlo simulations of this system. Now, we have two sets of samples: one from our model and one from the ...
9
votes
1answer
222 views
Closed form expression for the quantiles of $\alpha_1\sin(x)+\alpha_2\cos(x)$
I have two random variables, $\alpha_i\sim \text{iid }U(0,1),\;\;i=1,2$ where $U(0,1)$ is the
uniform 0-1 distribution.
Then, these yield a process, say:
$$P(x)=\alpha_1\sin(x)+\alpha_2\cos(x), ...
2
votes
1answer
118 views
Jointly stationary random process
If two wide-sense stationary processes $X(t)$ and $Y(t)$ are uncorrelated, then the cross correlation is
$R_{XY}(t_1,t_2) = E\{X(t_1)Y(t_2)\} = E\{X(t_1)\}E\{Y(t_2)\}$,
which will be a constant, ...
6
votes
2answers
2k views
How do betting houses determine betting odds for sports?
Let's take football (soccer) for example. There are 3 possible outcomes, home win, draw, away win. I took a random game from bet365
...
4
votes
2answers
485 views
Simulating a Gaussian process with an exponentially decaying covariance function
I'm trying to generate many draws (i.e., realizations) of
a Gaussian process $e_i(t)$, $1\leq t \leq T$ with mean 0 and covariance
function $\gamma(s,t)=\exp(-|t-s|)$.
Is there an efficient way ...
2
votes
0answers
81 views
Is there a version of the arcsine rule for biased coins?
I'm just reading about the arcsine rule for the first time in the context of coin flipping, i.e., that one player is very likely to lead a large percentage of the time in a finite number of trials.
...
1
vote
1answer
108 views
Perturbations of rational numbers with random variables
For $a_1,a_2$ and $b_1,b_2$, $\in\Re^+$, if $a_1<b_1$ , then for any perturbation by a random variable $\epsilon\in \Re^+$,
$$r_1=\frac{a_1+\epsilon}{b_1+\epsilon}>\frac{a_1}{b_1} $$ and if ...
2
votes
1answer
269 views
Expected hitting time of a level $a$ for Brownian motion
Let $\{W_t,t \geq 0\}$ be a standard Brownian motion under $\mathbb P$. Let $T_a$ be the hitting time of level $a$, that is:
$$T_a= \text{inf}\{t \geq 0:W_t=a\}.$$
From a proposition, we know that
...
9
votes
3answers
2k views
Expected number of coin tosses to get N consecutive, given M consecutive
Interviewstreet had their second CodeSprint in January that included the question below. The programmatic answer is posted but doesn't include a statistical explanation.
(You can see the original ...
5
votes
1answer
181 views
Taking random items out of a container with replacement
I take vitamins in the morning, but one of them I only take a half tablet.
So, I have an initial container with 100 full tablets, and every morning I take out a random tablet. If it's a full ...
0
votes
0answers
175 views
Moments of function of Poisson process
Mirror thread on Mathoverflow.
(I'm new to Poisson processes, so please edit if my terminology is incorrect.)
This is a special case of a problem I'm working on; hoping for intuition that will ...
9
votes
5answers
937 views
Time taken to hit a pattern of heads and tails in a series of coin-tosses
Inspired by Peter Donnelly's talk at TED, in which he discusses how long it would take for a certain pattern to appear in a series of coin tosses, I created the following script in R. Given two ...
4
votes
1answer
365 views
Conditions for Central Limit Theorem for dependent sequences
Cumbersome technical assumptions (e.g., mixing properties) are used in the literature to prove Central Limit Theorems for dependent sequences. I sketched a proof that does not require any of these ...
0
votes
1answer
327 views
Steady state probabilities for a continuous-time Markov chain
I have a finite state and time-homogeneous continuous-time Markov chain (CTMC) which is not irreducible. Will steady state probabilities exist for this CTMC? How to prove this?
3
votes
1answer
187 views
Meaning of this expectation equation?
I was actually looking at this problem on slide 12. I will write it here briefly:
Problem:
Unknown number of people arriving in a fixed time period and my goal is to maximize my probability of ...
6
votes
1answer
278 views
Origin of strange formula for equilibrium standard deviation
In the paper
M. Avellaneda and J. H. Lee, Statistical arbitrage in the U.S. equities market, July 2008,
in the Appendix on page 46, how does he get equilibrium standard deviation as following:
...
4
votes
1answer
257 views
Why local martingale property is important in Girsanov theorem?
In Girsanov theorem, the change of probability measure variable $Z_t = \frac{dQ}{dP}|_{\mathcal{F}_t}$, why does it need to be a martingale with respect to measure $P$ for the change of measure ...
8
votes
7answers
1k views
How will studying “stochastic processes” help me as a statistician?
I wish to decide if I should take a course called "INTRODUCTION TO STOCHASTIC PROCESSES" which will be held next semester in my University.
I asked the lecturer how studying such a course would help ...
3
votes
2answers
307 views
Trees generated by multi-type branching processes in n steps
I am trying to develop some algorithm to compute probabilities in multi-type branching trees, and I doubt I am doing this right...
Let us consider a multi-type branching process with two types, ...
6
votes
3answers
713 views
The expected value of random variable on tosses of a coin
Came across an interesting problem today. You are given a coin and x money, you double money if you get heads and lose half if tails on any toss.
What is the expected value of your money in n tries
...
4
votes
2answers
252 views
What is the distribution of $\chi^n_k$?
$\chi^n_k=\sum_{i=1}^kx_i^n$ where $x_i$ are Gaussian variables and $n>2$?
