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

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34 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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0answers
87 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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48 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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0answers
49 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
35 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
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0answers
262 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
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1answer
33 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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2answers
54 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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0answers
85 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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0answers
40 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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49 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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0answers
32 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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0answers
67 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 ...
2
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1answer
26 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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0answers
26 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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0answers
31 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
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1answer
94 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
56 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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0answers
105 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 ...
2
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2answers
141 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
21 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 ...
3
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1answer
144 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
175 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
988 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
30 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
303 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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0answers
18 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
46 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
29 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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21 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
990 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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1answer
38 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
91 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 ...
1
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1answer
63 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
311 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
39 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
47 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
112 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 ...
0
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1answer
34 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
108 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
40 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
167 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 ...
4
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2answers
225 views

Monte Carlo estimator of the number of 1's in a very long binary sequence

Preface The question below is related to a problem I am working on, which requires counting the number of times a logic-valued function evaluates "TRUE" given an input value. The size of my input set ...
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1answer
169 views

Obtaining useful information from graph obtained via Monte-Carlo Simulations

I've been running Monte Carlo Simulations on some Matlab code and then plot the graph shown below. I was just wondering what useful information I could collect from this graph? Edit: fit ...
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1answer
45 views

How do we calculate when to shout for the optimal payoff?

For example: A non-dividend paying stock is currently priced at $20, and you hold a put that allows early exercise in 2 months and in 4 months. The option expires in 6 months. Volatility is 30%, and r ...
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1answer
124 views

Buffon's experiment with squares

Say, we'd like to make the Buffon's experiment but with squares instead of needles. Notation: $d$ is the distance between lines $b$ is the square side length $y$ is the distance from the center of ...
2
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2answers
76 views

How many simulations for a game?

Lately I was interested by Monte-Carlo simulations. I found many papers about this approach in the Internet but for now they are too hard for me. I just want to start understanding this method with ...
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0answers
38 views

Average over all positive functions on the unit interval whose Lebesgue integral is one

I want to average over all positive functions on the unit interval whose Lebesgue integral is one. Formally, I want to compute the mean of the following probability distribution defined over function ...
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1answer
35 views

Bias Method in monte carlo integration

This is from a proof in my monte carlo course. let $h$ be a smooth function, $T_n = h(\bar{X})$ $\mu = E(X)$ then by taylor expansion $E(T_n -\tau) = E[h(\bar{X} -h(\mu)] = E[\bar{X} - \mu]h'(\mu)+ ...
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1answer
116 views

Monte carlo estimation of maximum likelihood estimators

I'm interested in numerically finding the maximum likelihood estimator of a parameter $\theta$, as well as the confidence interval of this estimator. First I'll describe the method I've been trying, ...