Questions tagged [monte-carlo]

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

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75 views

Is integration in high dimensions hard?

Consider the problem of estimating the integral $\int_{[0,1]^d} {\rm d}^dx f(x)$ where $f : [0,1]^d \to [a,b]$, to within relative error $\epsilon > 0$. My intuition is that this is an extremely ...
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15 views

Calculate RTP value in slots machines using Monte carlo simulations

Could you explain me how to use Monte Carlo simulation to calculate RTP in slots. I have found this sentence on the internet but without any explanations. For obtained RTP Monte-Carlo simulation is ...
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51 views

$x_1^2+x_2^2+x_3^2+…x_{10}^2 \leq1$ Do I use the Mathematica QuasiMonteCarlo method wrongley?

I want to find the volume of the multisphere restricted by $x_1^2+x_2^2+x_3^2+...x_{10}^2 \leq1$, by using NIngtegrate and the QuasiMonteCarlo method. I start with dim = 10 with the expression ...
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27 views

A question on hypercubes and the central limit theorem

I was reading a book on Monte Carlo methods and now I'm trying to make sense of an excercise. At one point they say that according to the central limit theorem most of the points ${\bf x} \in [0,1]^d$ ...
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1answer
37 views

PDF of $Q$ Random Variable

Let $X\sim N(0,25)$, $Y\sim N(10,100)$, $Z\sim N(-10,50)$ and $Q=\tan^{-1}\left(\frac{Z}{\sqrt {X^2+Y^2}}\right)$ When I simulate $Q$ random variable with Monte Carlo method, I'm getting this ...
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1answer
21 views

Computing a “Double Conditional Distribution”

This question comes in the context of Gibbs sampling, and I have posted it on the Stats Stack Exchange. Let us say we are considering random vectors in $\mathbb{R}^2$ of the form $(X,Y)^T$, such that:...
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14 views

Correctness of the following expected value

Let's say we have two functions f(x) and g(x), and we want to calculate the integral of the sum $\int f(x) + g(x)dx$ (the integral is finite) To do that, we randomly sample f(x) and g(x) with a ...
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1answer
25 views

Estimating Catalan numbers using Monte Carlo method

This question regards the classical problem of estimating Catalan numbers by performing a random walk on a grid of $n\times n$ squares. I will dectribe the problem for those who are not familiar with ...
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12 views

Lognormal VaR vs. Normal VaR

I understand that the normal and lognormal VaR have different formulas. My question is can there be a general statement made i.e. is the normal VaR larger or smaller in general for example for ...
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1answer
26 views

Monte Carlo Sampling with non-uniform distributions?

I'm currently studying Monte Carlo sampling, referencing Veach's "Robust Monte Carlo Methods for Light Transport Simulation". On page 63, he writes: The idea of Monte Carlo integration is to ...
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Cointegration of the xt and yt and the distribution of the test statistic

So I'm trying to solve the following question: I've managed to answer the first part of the question (which involved explaining the whole process) and I know that the test isn't following a standard ...
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1answer
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Monte Carlo and Sampling

Suppose that we have a finite state space $E$ and a distribution $\pi:E \rightarrow (0,1)$ with $\pi(x) >0$. The idea behind Monte Carlo is that we generate a Markov chain $X=(X_n,n\in \mathbb{N})$ ...
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7 views

Lognormal distribution bounds on monte carlo simulation

As the lognormal distribution imposes bounds of attainable correlations as discussed in https://stats.stackexchange.com/questions/41734/attainable-correlations-for-lognormal-random-variables my ...
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1answer
73 views

Markov chain Monte Carlo with stopping time

I'm doing my thesis where I'm required to compute the numerical value in the following problem: Let $(X_t)$ be a continuous-time Markov chain such that $X_0 = a$ almost surely. The state space $V$ ...
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8 views

Monte Carlo Minimum Variance Hedge Ratio

So I was running a monte carlo simulation for two assets and a portfolio consisting of 1 quantity of the first asset and short a fraction x of the second asset to hedge, where the fraction is ...
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24 views

Ratio of Normal Distribution probabilities

I am aware that the CDF of normal distributions are highly complex, but I have the following question. I have two normally distributed variables, A and B, ($N(\mu_A,\sigma_A$ and $N(\mu_B,\sigma_B$)); ...
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1answer
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Minimum Variance Hedge Ratio and Risk capital

I understand that the minimum variance hedge ratio minimizes the second moment of the portfolios. My question is how is it related to the size of the risk capital (which is calculated as the Value at ...
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1answer
22 views

Steps to do a Monte Carlo simulation

I'm trying to do a Monte Carlo simulation but I'm lost in the process. The big question I want to answer is what's the probability I have to do a certain amount of work. In my solution I've already ...
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11 views

Monte Carlo estimate for PI

Two well-known ways of estimating $\pi$ with Monte Carlo are dart throwing and Buffon's needle. Is there a MC experiment to estimate $\sqrt{\pi}$ at the usual rate of convergence?
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Attainable correlation bounds of two log-normal random variables

McNeill et al. (2015) mention that the attainable correlation for two lognormal random variables are not between 1 and -1 as they are not of the same type. Now I was wondering since the minimum ...
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1answer
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Doubt computing multiple integrals by Monte Carlo method

The question is: Using Monte Carlo method, compute $\int_{0}^{1} \int_{-1}^{1} (x+y) dx dy.$ My resolution until now: Let $g = x+y$. $\Theta = E[g(U_{1}, ..., U_{n})]; U{1}, ..., U_{n}$ random ...
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1answer
20 views

Covariance of Function of Antithetic Random Variables

So I'm given two functions $f$ and $g$, which are bounded and increasing. I need to prove that $Cov[f(G_1),g(-G_1)] \leq 0 $, where $G_1$ is a standard normal random variable (meaning that $-G_1 \sim ...
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1answer
23 views

MCMC: Integral Approximation?

Let's say I want to approximate the following integral: $$I = \int_0^5 R(x)f(x)dx$$ where $R(x)$ is a Rayleigh distribution and $f(x)$ is some generic function. I generate $n$ samples using ...
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23 views

Diagonal convergence of empirical measures of empirical measures

Given a probability measure $\mu$ (with density if needed), and its empirical measures $(\mu_n)_n$ which converges to $\mu$ in distribution almost surely as $n\to \infty$, we know: for fixed $n$, the ...
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Monte-Carlo integration of Riemann-Stieltjes integrals

It isn't clear to me how to tweak Monte Carlo integration to evaluate Riemann-Stieltjes integrals. So I have the following integral over $S$ which is some compact subset of $\mathbb{R}^n$ where $n$ ...
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22 views

How can one compute the Shapley value using Monte Carlo?

I am working on an application for which I need to compute the Shapley value (using the first formula from the definition section on the wiki). Since I am dealing with a large number of players (...
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Can control variate variance reduction be used when correlations are unknown?

There is a technique for Monte Carlo variance reduction called Control Variate (https://en.wikipedia.org/wiki/Control_variates). The idea is to use another variable to reduce the variance of the ...
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1answer
25 views

Sampling with exponential tilting from a Weibull distribution

Given the Weibull distribution: $$F(x)=1-e^{-\theta x^\beta}$$ Is there a way to sample from the tilted distribution? First we find probability density: $$f(x)=\beta x^{\beta-1}\theta e^{-\theta x^\...
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31 views

How to find the probability error of a Monte Carlo algorithm

I have the following algorithm that determines if max A = max B where A and B are sets with each having $k\geq 2$ elements such that, either, A∩B=∅, or A∩B is the singleton{s} and s = maxA = maxB. <...
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Lognormal asymmetry impact on long/short Value at Risk

To examine the Value at Risk implications for a portfolio consisting of a spot and futures time series I have generated a 1-day monte carlo simulation. I was long in the spot and short in the future (...
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35 views

Monte carlo simulation confidence interval coverage

$Y_{i}=\beta x_{i}+\epsilon_{i}$ where $ \epsilon_{i} \sim N(0,\tau)\ x_i$ are covariates and the profile likelihood of $ \beta $ is $\ l_p(\beta) = \ l_p(\beta,\hat\tau(\beta)) =-\frac{n}{2}log\...
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4 views

Simulating Probabilistic Behaviour

I am trying to develop a code in the Mathematica program to simulate the problem below: I need to run 1,000,000 simulations to determine the mean score for throwing five darts I am having difficulty ...
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1answer
34 views

Algorithm for generating a sample for geometric distribution using a uniform distribution

Let U be uniformly distributed on the interval $(0,1)$ and $$Y=\frac{\log(1-U)}{\log(1+p)}+1.$$ Then compute $\Bbb{P}(Y=k)$ and use it to provide an algorithm for generating a sample for geometric ...
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Understanding acceptance-rejection method - more general point of view

I am searching for a proper geometrical explanation of the acceptance-rejection method. Usually it is presented in a way such that, to sample $X$ from a distribution $f(x)$ with $x\in(a,b)$, one ...
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Perfect Sampling - Reuse of random bits

I'm currently studying the Perfect Sampling approach to Markov Chain Monte Carlo proposed by Propp and Wilson in 1996.Though having understood most important aspects, I'm still struggling to figure ...
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30 views

Uncertainty estimate of a sample of one

A monte-carlo approach to a problem produces many thousands of possible solutions, each ranked according to the total residue from observed data it implies, so that the solution with the lowest net ...
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Can discrepancies of a sequence in the interval decay arbitrarily slowly?

So let's suppose we have a sequence of points in the interval: $(x_n)_{n=1}^\infty,\ x_n \in [0,1]$, which becomes equidistributed as $n \to \infty$: that is to say, for all $0\leq a<b \leq 1$, the ...
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Got stuck at trying to figure out what the single shot at inference for Variational Autoencoder should be

Let's say you have an already trained Variational Autoencoder where the parameters are $\phi, \theta$ for the recognition and generative models respectively. Let's also assume you have the following ...
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11 views

How Importance Sampling Works

I understand that importance sampling involves sampling from one distribution to estimate the expected value of another. We do this when the distribution of the random variable whose mean we are after ...
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23 views

Discrepancy and Dispersion of a perfect square

Consider the unit square $[0,1]^2$. Let $R$ be the family of all axis parallel rectangles inside $[0,1]^2$ (not necessarily anchored at the origin). Suppose $n$ is a perfect square. Let $\{P = (\frac{...
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What is the point in using Monte Carlo simulation to compute π?

I recently read about computing π using a Monte Carlo simulation in a MIT Course, and I thought that π is in no way related to randomness, absolutely it is related to the Pythagorean Theorem, ...
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34 views

Is there a formula for $\mu\left(\left\{w>0\right\}\right)$ if $w$ is a $[0,1]$-valued function?

Let $(E,\mathcal E,\lambda)$ be a measure space $I$ be a finite nonempty set $p,q_i$ be probability densities on $(E,\mathcal E,\lambda)$ for $i\in I$ $\mu:=p\lambda$ $w_i:E\to[0,1]$ be $\mathcal E$-...
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23 views

If $\cup_iA_i=A\subseteq\{\sum_iw_i=1\}$, can we show that $\sum_i\left(\int w_i\:{\rm d}μ\right)\int_{A_i}g\:{\rm d}μ=\int_Ag\:{\rm d}μ$?

Let $(E,\mathcal E,\lambda)$ be a measure space $I$ be a finite nonempty set $p,q_i$ be probability densities on $(E,\mathcal E,\lambda)$ for $i\in I$ $\mu:=p\lambda$ $w_i:E\to[0,1]$ be $\mathcal E$-...
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How to partition a probability distribution function to find expected value?

Given a probability distribution function $p(x)$, the corresponding cumulative distribution function $F(x)$ and $y(x)$ which is computationally expensive to evaluate, I would like to find the expected ...
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2answers
45 views

Evaluating a double integral with monte carlo integration?

I am trying to find an equation that estimates the following integral: $$\int_0^1 \int_0^1 e^{(x+y)^2} \,dx\,dy$$ where I am given a list of different uniform RV's $m_1,m_2,\ldots$ and $n_1,n_2,\...
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Choosing a discrete proposal distribution for a multinomial process in a particle filter.

I have a particle filter and I know the target distribution is modelled via the multinomial distribution. I had a very similar multinomial distribution for my proposal distribution. However, I ended ...
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1answer
57 views

Probability that one Gaussian RV exceeds all others

Imagine we have $k$ Gaussian RVs $$ X_i \sim N(\mu_i, \sigma_i^2) \text{ for } i=1, \ldots, k $$ and we sample from each of them independently to produce a vector, $\vec{x} = (x_1, \ldots, x_k)$. For ...
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46 views

Geometric intuition behind Monte Carlo integration

I found 2 seemingly different explanations of the geometric intuition behind Monte Carlo integration. Watch from 4:51 of this video by Jared Niemi. https://youtu.be/MKnjsqYVG4Y There is a set of 4 ...
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1answer
43 views

Plotting convergence of Monte Carlo estimation of Pi to display convergence rate

The answer to this thread states the variance for 2 Monte Carlo estimators of $\pi$. Variance of Area and Average Estimators in Monte Carlo Estimation of Pi The variance for both methods is ...
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1answer
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Variance of Area and Average Estimators in Monte Carlo Estimation of Pi

I know of 2 Monte Carlo estimators of $\pi$. Rick Wicklin discusses these 2 methods here. https://blogs.sas.com/content/iml/2016/03/14/monte-carlo-estimates-of-pi.html 1) The area method throws ...

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