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

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Processing Issues [on hold]

I have to run heaps of models using Monte Carlo simulation so in essence it takes forever. Are there any websites that I can rent processing time so that I can get it done really quickly? Cheers for ...
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14 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 ...
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1answer
16 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 ...
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27 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 ...
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22 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), ...
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43 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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27 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 $$ ...
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47 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 ...
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1answer
24 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 ...
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8 views

cholesky decomposition and generate random vector

Hi someone please explain me this code : it is about running stress scenario simulation with two hypothetical shocks on macroecomics variables (gdpgap- exchange rate). The first hypothetical ...
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2answers
47 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 ...
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21 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 ...
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26 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 ...
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28 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 ...
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42 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 ...
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12 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, ...
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30 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 ...
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9 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 ...
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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 ...
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25 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 ...
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1answer
63 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 ...
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1answer
43 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 ...
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58 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 ...
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2answers
109 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 ...
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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 ...
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1answer
76 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: ...
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1answer
91 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 ...
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1answer
154 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; ...
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1answer
22 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
89 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 ...
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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 ...
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1answer
32 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 ...
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2answers
28 views

Let Y be a random variable with $0\le Y\le 1.$ [duplicate]

Let Y be a random variable with $$0\le Y\le 1.$$Show that $$var(Y)\le 1/4 $$ and that $$var(Y)= 1/4 $$ if and only if P(0)=1/2=P(1).
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19 views

How to show that the variance increases with the dimension $n$?

This can be seen as a statistics related question, but it is actually a more general mathematics related one. I am trying to understand the Particle Filter and the motivation to use it over the ...
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3answers
124 views

Using loop to approximate pi (Monte Carlo, MATLAB)

I've written the following code, based on a for loop to approximate the number pi using the Monte-Carlo-method for 100, 1000, 10000 and 100000 random points. ...
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20 views

Monte carlo error: Combining “experimental” and statistical errors

I'm doing a slightly involved Monte Carlo approximation of a quantity $E$ where I end up with the following formula: $E=\frac{\sum_{i=1}^np_ie_iG_i}{\sum_{i=1}^np_iG_i}\ .\ \ \ \ \ \ \ \ \ $ (1) ...
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1answer
26 views

Generating two $-1$ correlated Poisson random variables with parameter $5$

Is it possible to generate two random variables $X$ and $Y$ that are both $Poisson(5)$ with $Corr(X,Y)=-1$? Why? I was thinking about generating $3$ independent Poisson random variables $Z_1,Z_2, and ...
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1answer
69 views

How to mathematically prove that we are sampling from same distributions?

The content of this question is about rigorously proving something which is otherwise considered easily correct intuitively. Let's assume we have a multivariate distribution $g(x_1,x_2,...,x_n)$ over ...
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1answer
39 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 ...
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2answers
132 views

variance of 26 cards chosen from a deck

Suppose I have a well shuffled deck and I am trying to find the variance of 26 cards randomly chosen without replacement from a deck, assuming the values are from 1 to 13 for the cards. Since the mean ...
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0answers
31 views

Monte Carlo with error on individual samples

I'm performing a Monte Carlo integration where the individual samples have an error, and I'm wondering how to estimate the final error. Some more detail: The integral E I'm after is estimated in the ...
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1answer
34 views

Calculate expectation under risk neutral measure: $\mathbb{E_Q}(\max(S-1,0))$

I am busy with a numerical simulation and I want the calculate the following expectation under the risk neutral measure: $\mathbb{E_Q}(\max(S-1,0))$. $S$ is some variable that I calculated using ...
2
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1answer
80 views

Proving that Markov Chain Monte Carlo converges

I am trying to understand how the very basic Markov Chain Monte Carlo approach works: We try to approximately calculate the expected value $E_{\pi(x)}[X]$ by drawing sequential samples from a Markov ...
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0answers
30 views

Monte carlo formula to compute the approximation of variance of MLE

In the book of "Monte Carlo Statistical Methods", the book gives an approximation formula for the variance of MLE, Later on, the book mentions that this approximation formula can be written as ...
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10 views

control variates - estimating definite integrals

What are the good techniques to find $g(x)$ so that $f (x) - g(x)$ is minimal, in order to evaluate $\int_a^b [(f(x) - g(x)) + g(x)]dx$?
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1answer
32 views

Analytic approach to find probability and total value of a set of independent events

I have a forecasting worksheet which describes a set (worksheet) of independent events, all of which have a likelihood of happening given as a probability (e.g. 0.7). Every event also has a yield ...
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1answer
84 views

Sample uniform direction within cone

My question is pretty much the same as this question below, however I came up with a potential solution to this problem that I didn't see an answer to in the other question and I was wondering if it ...
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1answer
34 views

Determining Errors in Monte Carlo Simulation

I was wondering if anyone could throw light on possible errors associated with Monte Carlo sampling. I seem to be getting values that are slightly different each time despite running my model for ...
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2answers
112 views

Algorithm to find best in class of groups with weighting?

I have widgets and a single widget will have attributes of: Name Weight (decimal from 0-1) Group (letter A-F) Price (an integer from 1 - 100) I must pick one ...
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23 views

Antithetic pair of non-independent normal random variables

Suppose that I have two non-independent normal random variables, X and Y such that $(X,Y)$ has mean 0 and the following variance covariance matrix: \begin{bmatrix} 1 & \rho ...