Questions on Monte Carlo methods, methods that require the repeated generation of (pseudo-, quasi-)random numbers for computing their results.

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2
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1answer
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Applying MCMC Metropolis algorithm

I'm interested in all possible paths (on the grid $\mathbb{N}^2 $) that goes from $ (0,0) $ to $ (n, n) $. At each step there are two possibilities: go right or go up. The path is a sequence $ ...
0
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2answers
46 views

How to use Monte Carlo method to find volume?

So the question asks to use the Monte Carlo method to find the volume of an irregular figure defined as: ...
-1
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2answers
86 views

Are irreducible, positiv-definite Markov chains aperiodic?

If $M$ is the transition matrix of a discrete Markov chain, and $M$ is both irreducible, symmetric and positiv-definite, is the resulting Markov chain necessarily aperiodic? In my intuition, ...
-1
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0answers
12 views

Rejection sampling multivariate case

Suppose $f$ is a complicated pdf of a random variable $X$ from which it is not easy to generate random samples. On the other hand, suppose $g$ is a pdf from which it is easy to generate random ...
-2
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0answers
16 views

Manifolds and Random Number Generators [closed]

I was reading this answer on quora: http://www.quora.com/What-are-the-most-important-uses-for-randomness/answer/Subit-Chakrabarti and was wondering about the following passage: Of course, a much ...
0
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1answer
15 views

Random sampling using Metropolis vs. Accept-Reject

I'm working on a project comparing the accept-reject (von Neumann) and Metropolis methods of sampling. I'm generating a sample of size $N$ from a normal distribution $N(1,(\frac{1}{2})^2)$. What I ...
0
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0answers
16 views

How to rewrite function for squared uniform distribution

The question is as follows: I am evaluating the following integral: $$\int_o^1\frac{\exp(\sqrt{1-x^2})}{\sqrt{x}}dx$$ by assuming it equals $E[f(U)]$ for a uniform distribution. I worked it out via ...
0
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1answer
11 views

Logic with increasing monte carlo possible output

I am working on Monte Carlo algorithm : Given that you have an experiment MC which has a p-correct of 75%, which means it gives you the right answer 75% of the time. You run MC3 which repeats MC 3 ...
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0answers
12 views

SEnsitivity Indices are non zero

I am trying to compute the sensitivity indices (SI) of a function using Monte Carlo simulation. I had written a matlab code that perform the computation directly and just return the final answer of my ...
2
votes
1answer
23 views

Does the perimeter of a 2-D object “count” toward its area?

I'm writing a quick Monte-Carlo simulation for a class in Matlab in order to estimate the value of pi as demonstrated in this gif: http://en.wikipedia.org/wiki/File:Pi_30K.gif However, I'm not sure ...
0
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0answers
18 views

Is there a such thing as a quasi-random shuffle?

I've recently experimented with Quasi-random numbers in monte-carlo applications. Is there a way to construct a quasi-random shuffle? By that I mean can I take a sequence $Q$ and shuffle it to produce ...
2
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0answers
9 views

Monte Carlo Markov Chain Simulation Issues

The Markov Chain is uniformly distributed across all $50$x$50$ matrices of entries $0$ and $$1 with no neighboring $1's$. I am supposed to run a MC simulation to check the probability that the ...
0
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0answers
28 views

Monte Carlo Markov Chain simulation

I am going to post the python code logic we used however I want someone to look at the number that are printing out. The Markov chain is uniformly distributed across all $50x50$ matrices with entries ...
13
votes
3answers
767 views

Why does Monte-Carlo integration work better than naive numerical integration in high dimensions?

Can anyone explain simply why Monte-Carlo works better than naive Riemann integration in high dimensions? I do not understand how chosing randomly the points on which you evaluate the function can ...
0
votes
0answers
8 views

Using the rejection method to generate values

I'm trying to solve the following problem on rejection sampling: I think I have a good idea about what rectangle I should be using. In my mind, it would be a rectangle just large enough to encompass ...
0
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0answers
7 views

Sensitivity analysis of paramaters and input variables

I am trying to perform a sensitivity analysis of an optimization problem $f(x,\alpha)= \min_{ Q} {g(x,\alpha , Q)}$ where $x$ is an input variable for our function, and $\alpha $ is a parameter. ...
0
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1answer
23 views

Compute the Std. Deviation of Multiple Monte Carlo Estimation of $\pi$

For a school programming assignment, I am trying to compute the value of $\pi$ via the classic Monte Carlo estimation of $\pi$. In the experiment, we throw a variable number of darts at a circle that ...
4
votes
0answers
404 views

Simulating from a Multivariate Gaussian without Cholesky

I'd like to draw a sample from a multivariate Gaussian distribution $\mathcal{N}(\mu, \Sigma)$, where $\mu$ is the mean vector (can assume it to be $\boldsymbol{0}$), and $\Sigma$ is a sparse positive ...
0
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1answer
36 views

Uniform convergence of Monte Carlo approximation

Usually Monte Carlo method is used to compute integration. For example, let $g(x,\theta)$ be a continuous function about $x$ and $\theta$, $f(x \mid \theta)$ is a continuous pdf with parameter ...
0
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0answers
15 views

How to sample points from a bounded polytope?

I have a bounded polytope $C \subset \mathbb{R}^n$ characterized by the following restraints: $$ x \in C \Leftrightarrow \sum_{i=1}^n x_i = 1 \text{ and } Ax \leq b$$ for some matrix $A \in ...
0
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1answer
26 views

Approximation and Monte Carlo simulation.

I am a bit up over my head here, I will present an argument and then I hope you guys will say if my reasoning is correct or what should be changed, ultimately I am hoping to say something qualified ...
0
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0answers
52 views

Correlated variables from Latin Hypercube

Say I have a vector $\mathbf{Y}$ of $n$ normally distributed random variables. I have its mean vector $\mu$ and covariance matrix $\Sigma$. Normally if I were to generate a sample, I would decompose ...
3
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0answers
25 views

How can I prove that the error of the Monte Carlo method for finding $\pi$ decreases as $N$, the number of samples, increases?

I am currently playing around with the Monte Carlo method of finding $\pi$. The idea is pretty simple, I work with a unit square of length one on each side, with the coordinates of $(0,0) , (1,0), ...
1
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0answers
44 views

Linear Filtering Problem (Keynman Fac/Particle Model)

$lienar Filtering Problem $$X_n^1 = X_{n-1}^1 + \epsilon_n *W_n $$ $$X_n^2 = (1-\alpha* \delta) X_{n-1}^2 + \beta*\delta X_n^1 $$ $$X_n^3 = X_{n-1}^3 + \delta*X_n^2$$ above is $$\approx$$ $$dX_n^1 ...
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0answers
33 views

Normalisation of Monte Carlo overlap

Background: In quantum mechanics you are sometimes required to compute the overlap (inner product) of two wave functions (square integrable complex functions) $\Psi(x)$ and $\Phi_i(x)$ as $$ ...
2
votes
1answer
90 views

Using Monte Carlo method to estimate the value of $\int_0^1\int_0^1e^{xy} \ dx \ dy$

Monte Carlo, estimate the value of $$\int_0^1\int_0^1e^{xy} \ dx \ dy.$$ I am using matlab to solve this problem. My code is the following: ...
2
votes
0answers
121 views

Law of a geometric brownian motion first hitting time (proof checking)

I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all subsequent simulation. Could someone ...
1
vote
1answer
27 views

Find Monte Carlo Variance When Expected Value is not Known

I'm working on a problem that can be approached in two different ways. Both are Monte Carlo algorithms--but it's a hard problem, so I am unsure whether the expected values are indeed the same. I ...
2
votes
2answers
50 views

Sampling from the diamond: $|x_1|+\ldots+|x_n| \le 1$?

Let $\left(x_1, \ldots, x_n \right)$ be a point in $\mathbb R^n$. Sample uniformly at random from the diamond $$ |x_1|+\ldots+|x_n| \le 1. $$ In $\mathbb R^2$, one way is to sample the square, then ...
0
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0answers
30 views

Optimizing Buffon's needle to minimize the variance in $\pi$

This is a homework problem so I don't expect a full answer, I'm looking for some pointers on where to start. Problem text: Find L (stick length) and D (separation between lines) that minimizes the ...
0
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0answers
39 views

Calculating error bounds for linear regression fit using Monte Carlo methods

so my question is pretty simple. I have some data that has a known error in the y coordinate and I'm fitting it to a linear model using least-squares. Now normally I know we neglect the error and ...
0
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0answers
33 views

Multi-armed bandit optimization

I've got a variation of multi-armed bandit problem, where I need not to minimize the regret, but to find a bandit with a maximum reward. Could you please tell how this problem is called or suggest ...
0
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0answers
44 views

Sampling and averaging in Monte Carlo Simulation

(First of all, I apologize for the vague title. Couldn't think of rather proper one.) Let's say that we have 10 items where each item has probability distribution of one's own, say Lognormal ...
2
votes
0answers
17 views

Finding Volume of Monte Carlo Integration

Suppose $\mathbf{X}\in\mathrm{R}^n$ is an $n-$ dimensional random vector having joint Gaussian distribution i.e. $\mathbf{X}\sim\mathcal{N}\left(\boldsymbol\mu,\boldsymbol\Sigma\right)$, where, ...
19
votes
2answers
5k views

Probability that a stick randomly broken in two places can form a triangle

Randomly break a stick (or a piece of dry spaghetti, etc.) in two places, forming three pieces. The probability that these three pieces can form a triangle is $\frac14$ (coordinatize the stick form ...
1
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0answers
37 views

Convergence of Monte Carlo simulation

I am not sure if this is a valid question but here goes. For the monte carlo method I know that estimation of the mean is also a random quantity and follows a normal distribution. The standard error ...
0
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0answers
11 views

Metropolis Monte Carlo with modified acceptance

What happens, if I change the acceptance criterion in a Metropolis Monte Carlo algorithm? I do know the classic proof of detailed balance, which for symmetric transition matrices gives a set of ...
0
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0answers
19 views

Simultaneous multiple perturbations in Markov chain Monte Carlo

I'm coding a McMC algorithm for geophysical applications. Using the Metropolis-Hastings scheme to accept/reject the proposed models is smth that i thought i completely understood, but i don't. To be ...
0
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0answers
27 views

Correlating random numbers seems to skew the data

I am trying to generate a series of correlated random numbers that represent currency exchange rates for a Monte-Carlo simulation. I am attempting to do this via a Cholesky decomposition of the ...
0
votes
1answer
74 views

Forecasting future revenue and expensces

I am trying to forecast future revenue and expenses in a company. In the past I used moving average method but later I am more inclined to try to do that by using monte carlo simulation. I am ...
1
vote
1answer
47 views

Numerical CDF estimation for complicated random variable

Given a combination $U$ of several random variables $X,Y,Z...$ with known distributions, what is an efficient numerical algorithm to estimate PDF or CDF of $U$, if its CDF, PDF, characteristic ...
2
votes
2answers
115 views

Monte Carlo Importance Sampling

I am reading the book on Monte Carlo by Sobol (A Primer for the Monte Carlo Method). In the section on Importance Sampling, he writes: $I = \int_a^b g(x) \: dx$ "to compute this integral, we could ...
1
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1answer
49 views

Resolve integral with importance sample Monte Carlo

I'm trying to compute the integral $$\int_{a}^{b}(\sin( 1 + x ) + \cos( 1 + x ))e^{-x}\ dx$$ using importance sample Monte Carlo method. The exercise ask to use Cauchy Distribution to resolve the ...
1
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0answers
68 views

What does “sequence is equidistributed in [0, 2]” mean?

I was reading an article in which they are mentioning this sentence: "sequence is equidistributed in [0, 2]" where the sequence in question, is a sequence of real number (the article in question is ...
0
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1answer
19 views

Monte Carlo Methods for non-orthogonal functions

I'm trying to approximate a function using a set of piecewise polynomials. For example, perhaps I'd like to uniformly split the domain [-1,+1] 20 times and place Wendland RBF, Gaussian, or maybe a ...
0
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1answer
115 views

Double Integrals & Expected Value Monte Carlo Method

Tell me if I'm wrong Let $\Omega = [a,b]\times[c,d]\subseteq\mathbb{R}^2$, then $$ \iint_\Omega ...
0
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1answer
365 views

Evaluating double integrals using monte carlo methods in matlab.

I used the monte carlo method to integrate $\int_{0}^{1}x^2dx$ in matlab. My matlab code was simply the following: A=1; N=10000; s=0; for i=1:N x=rand; y=rand; if y<= x^2; ...
1
vote
1answer
25 views

Understanding claim in Newman and Barkema's Monte Carlo book

In Newman and Barkema's Monte Carlo Methods in mathematical physics, on page 23-24, the following claim is made: "Assume we have a function f(x) and the integral $I(x)=\int_0^xf(x')dx'$. Then pick a ...
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0answers
219 views

Monte Carlo standard deviation of the mean estimate too small.

I'm doing a Monte Carlo calculation and use the standard deviation of the mean $\sigma_M$ as the error. To get an estimate of this from the regular standard deviation I use $$\sigma_M=\dfrac\sigma ...
0
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0answers
17 views

Showing that the variance increases with the dimension of the random vector

This is actually related to a more complex question; but I want to re-ask it by trying to simplifying it as possible: 1- We have $n$ dimensional functions of the form $f_n:\mathbb{R}^{n} \mapsto ...