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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Is the sample path from Monte Carlo Simulation follows CLT?

I'm pricing some derivative securities with Monte Carlo Simulation. I don't know population follows which distribution. But When I some calculation with my samples(all the each path) ,for example $X = ...
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Monte Carlo simulation at football

Would you please help and clarify how to run a Monte Carlo simulation to determine the probability of one of the two teams winning a match given the average number of goals for each team (in matches ...
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Monte Carlo estimation of this probability

Let $p=P(X+Y\geq t)$ and $t\in \Bbb R$. Question: Using the classic Monte Carlo method, find an estimator $p_n$ of $p$ using $F^{-1}_X$ and $F^{-1}_Y$ Attempt: I defined $$Z=X+Y$$ then I expressed $$p=...
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Monte Carlo computation of probability of a subset of samples

I would like to compute the probability for some subset $\omega \subset \Omega$ of events to occur, i.e. $P(\omega) = \sum_{x \in \omega} P(x)$ where I know $P(x)$ for all $x \in \Omega$, which are ...
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Why does this plotted data dip before rising?

The red plot shows the amount of total time spent waiting at a given trade frequency. The data was generated by continually letting an asset price go up and down randomly, then repeatedly waiting for ...
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Expected Value of Slot Machine

I have a probability question that has to do with slot machines. Here is how the game works: There are two reels, one on the left and one on the right. There are several symbols on each reel, one of ...
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Probability of a point sampled from a ball lying in a spherical cap/segment

What is the probability that a point sampled from an n-ball lies in a spherical segment in the ball? Alternatively, what is the probability that a point sampled from an n-ball lies in a spherical cap ...
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Importance Sampling: What is the role of the target distribution $p(x)$ in Monte Carlo integration?

I am currently implementing a few different approaches to Monte Carlo Integration and there is a conceptual hurdle that I don’t fully understand yet. The notation varies from one source to another, ...
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Volume of a spherical segment in high dimensions

Consider 2 parallel hyperplanes of the type $0 \leq \langle w,x \rangle + b$ and $\langle w,x \rangle + b < c$ where $x, w \in \mathbb{R}^n ; b,c \in \mathbb{R}$ cutting an $n$-ball with radius $r$ ...
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Metropolis-Hastings : Finding the transition matrix given the state space and the stationary distribution

In this paper Diaconis, P. (2009). The markov chain monte carlo revolution. Bulletin of the American Mathematical Society, 46(2), 179-205. https://math.uchicago.edu/~shmuel/Network-course-readings/...
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How to use Monte Carlo method to calculate the volume of a cone inscribed in the unit cube?

unit cube L=1 how do i use random numbers with monte carlo method to calculate the volume of this cone? if anyone can use matlab to solve it, i appreciate it!
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Modeling dependent events

Say we have event A (saving a life) and event B (person living after initially getting saved). A and B are dependent events as p(A|B) = 1. I want to model the probability of a person living which is p(...
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What is the probability that a Markov chain transitions between states if it passes through a specified intermediate transition?

Consider a discrete-time finite Markov chain with transition probability matrix $P$. One of the foundational results of Markov theory is, of course, that the probability that the chain transitions ...
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Metropolis-Hastings algorithm Hastings ratio

This question is probably elementary, but it is not obvious to me how to compute the Hastings ratio since there is p in it which is the very target distribution we would like to find.
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Box-Muller Transformation: Polar Coordinates Interpretation

I am aware that the Box-Muller transform leverages polar coordinates to arrive at the final transformations by plotting two uniform random variables, $(u, v)$ in the Cartesian plane. I have not seen ...
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Does there exist a uniform Monte-Carlo Approximation of certain function classes?

Given a measurable and bounded function $f:X \to \mathbb{R}$ on a metric-measure space $(X, d, \mathcal{P})$, we can approximate $\int_\chi f$ in terms of $N$ iid samples $X_1, \ldots, X_N \sim \...
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Estimate of Expectation of Estimated Function

Let $X_1,X_2,...$ be a sequence of random values distributed uniformly in $[0,1]$. By the law of large numbers, we have for any integrable function $f=f(x)$ (convergence in probability to an ...
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Monte Carlo integration - Intuition

I am studying engineering and I am learning about the Monte Carlo integration which I find a bit hard to grasp the maths. I have watched a few videos and read some other notes and I feel I understand ...
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Convergence of a type of Monte Carlo integration which is different from the common one

$a_i \in (0,1)$, $I = 1,\cdots,N$ are $N$ random samples from uniform distribution $U(0,1)$. $a_i$ is in ascending order $a_1 < a_2 < \cdots < a_N$. $Q(p)$, $p\in(0,1)$ is a differentiable ...
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Singular Value Decomposition (SVD) with Monotonic Constraint

I am trying to compress some cumulative distribution functions (CDFs) which are stored in an $N \times M$ matrix $A$. Each of the $M$ columns contains $N$ monotonically-increasing values which might ...
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Monte Carlo conditional variance reduction

On this lecture at page 9: https://home.csulb.edu/~tebert/teaching/lectures/552/vr/vr.pdf you can found: $$\operatorname{Var}\left(1-U^{1 / 2}\right)=E\left[\left(1-U^{2}\right)\right]-\left(\frac{\pi}...
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Monte Carlo Tree Search

You are to manually run the MCTS algo- rithm for the navigation example covered in class for 10 iterations. For this, you will need simulation results for nodes at tree depth up to 2, which is ...
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Finding Monte Carlo estimate $\hat{K}$ using Monte Carlo integration

Let $$f\left(x\right)=K\left[sin^2\left(6x\right)+3cos^2\left(x\right)sin^2\left(4x\right)+1\right]e^{-\frac{x^2}{2}},\:-\infty <x<\infty $$ be the probability density function of a random ...
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Partition in Gibbs Sampling; obtaining the conditional Posterior Distribution

I would like to know the conditional posterior distribution if the following equation is partitioned by the unknowns by $ [\mu, \sigma^{2}]$ and $x_{1}$ of the equation: $$ \begin{aligned} g\left(\mu,...
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Proof of convergence for the Monte-Carlo method in the Linear Transport Equation

There exists a well known procedure for simulating the transport equation using the Monte Carlo (MC) approach. I want to understand the mathematical connection (not physical) of this approach to the ...
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Feynman-Kac proof

I have seen numerous proofs, showing that if $u\left(x,t\right)$ satisfies a PDE of the form: $$\frac{\partial u}{\partial t}+\mu\left(x,t\right) \frac{\partial u}{\partial x}+\frac{1}{2}{\left(\sigma\...
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quasi-Newton methods with Monte Carlo sampling

I have an optimisation problem of the form $$\text{argmax}_{\theta} \ L(\theta),$$ where $$L(\theta) = \ \mathbb{E}_{X\sim p(\cdot)}\left[f(X,\theta)\right],$$ and where $p(\cdot)$ is a distribution ...
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Does Metropolis-Hastings make sence if I have the normalizing constant of $p(x)$?

I have $x \sim N(\mu, diag(\sigma))$ and a non-trivial function $y = f(x)$ which transforms the distribution. I want to approximate the distribution of $y$ by a normal distribution and thought the ...
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Training neural networks | Monte Carlo Methods

Assume that we have a dataset D = {(x_1, y_1), ... , (x_n, y_n)}. We want to train a neural network and update the weights with gradients computed on every mini-batch. If we shuffle our mini-batches ...
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How is Simulated Annealing an "Adaptation of The Metropolis-Hastings Algorithm"?

I'm trying to understand how simulated annealing is related to the Metropolis-Hastings algorithm, that is, if they are at all. The Wikipedia page states that simulated annealing "is an adaptation ...
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Conditional Variance Implementation, Integral approximation with Monte Carlo

as part of my master's thesis, I have to implement the following conditional variance \begin{align*} \mathbb{V}ar[\hat{v}|\hat{y}] &= \mathbb{E}\left[\hat{v}^{2} \mid \hat{y}\right] - \mathbb{E}\...
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Finding the approximate value of improper integral using the Monte-Carlo method

Find the approximate value of the improper integral $$ \int_{-3}^{\infty} \left( \int_{0.5}^{\infty} \frac{2+\sin(x+y)}{e^{0.4x}+0.4y^2} \,dx \right) \, dy $$ using the Monte-Carlo method. I have ...
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Intuition behind using random variables in Monte Carlo methods / localization

I am trying to get a better intuitive understanding of why Monte Carlo works so well in approximating a solution to complex problems, such as calculating irrational numbers or the Particle Filter / ...
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$f(X) = \mathbb{E}[Y|X]$ maximizes correlation coefficient $Cor(Y,f(X))$?

Given two random variables $X$ and $Y$, I want to find a function $f(\cdot)$ such that the correlation coefficient $Cor(Y,f(X)) = \frac{Cov(Y,f(X))}{Std(Y)Std(f(X))}$ is maximized. Intuitively, such $...
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The question of standard deviation in gap statistics

From the paper of gap statistics, $\text { we estimate } E_{n}^{*}\left\{\log \left(W_{k}\right)\right\} \text { by an average of } B \text { copies } \log \left(W_{k}^{*}\right)$, $sd(k)$ is the ...
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3 votes
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Construction of Markov Chain

Let's consider a toy example of applying a Markov Chain Monte Carlo (MCMC) method. Let $X$ be a discrete random variable whose unnormalized probability mass function is $$ \tilde{p}(X=1)=4, \tilde{p}(...
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Prove the existence of limiting distribution in MCMC.

I know the MCMC can produce a Markov chain with desired stationary distribution, which can be proved from the Detailed Balance Condition πiP_ij=πjP_ji. But I don't know how to ensure the existence of ...
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Multivariate control variate technique

In the control variate technique, one tries to improve convergence of the expected value of a random variable $X$, estimated from simulating a range of $n$ Monte Carlo simulations. This is done by ...
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1 answer
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Understanding the Metropolis-Hastings Algorithm

Can someone please tell me if I have correctly written the code for MCMC Sampling (Metropolis Hastings Algorithm)? Given a multivariate joint probability distribution function, I am interested in ...
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An explanation for drift in diffusion processes in terms of probability distributions?

Consider the following Python implementation of an empirical experiment involving a diffusion processes. ...
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monte-carlo gone wrong

Can someone give examples where defective pseudo-random number generators gave incorrect simulation (or sampling, or polling, or gaming, …) results? I am aware of the RANDU debacle and of Ferrenberg, ...
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Producing direction unit vectors from 2D angle Gaussian distribution

I am trying to use the model given in the paper: Representation of a Gaussian beam by rays https://aapt.scitation.org/doi/10.1119/1.2201857 The model says that initial ray positions are normally ...
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Choosing m, with replacement, from n, with a probability twist

I need to be able to answer questions like the one shown below. I believe this is related to the Monte-Carlo simulation: A set holds integers $1$ to $1000.$ How many must I randomly choose, with ...
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Laplace transform / gaussian random variable

We have a gaussian random variable $X \sim N(0, \sigma^{2})$, with $\sigma^{2}$ unknown. the Laplace transform given by: $\phi(t) := \mathbb{E} [e^{tX}]$ = $e^{{\sigma}^{2} t^2/2}$ I need to make i.i....
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Monte Carlo for estimating small

Suppose that I have two samples $x_1,\cdots,x_n$ and $y_1,\cdots, y_m$ of two random variables $X$ and $Y$, with $m$ and $n$ very large. I want to estimate the probability $P(Y>X)$ which I know is ...
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2 votes
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Numerical Integration of a high dimensional function

My problem is the following, I have a function, with an input vector of $\mathbb{R^{3072}}$, which outputs {$\mathbb{ x \in R : 0 \leq x \leq 1}$}. I want to find the integral of the function over the ...
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Simple Monte-Carlo results

In a formula where R= (w1X1+w2X2) * (W3X3+ W4X4) and 0<=w<=1, 0<=X<=10 I want to use Monte-Carlo to generate results. I am setting the formula in excel, where cells containing W,X are ...
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Monte Carlo and Importance Sampling

So I am messing around with Monte Carlo Integration and trying to reduce variance thru Importance Sampling. The Monte Carlo I am considering is: $$ F^N = \displaystyle \frac{1}{N} \sum_{i=1}^{N}\frac{...
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Balls into bins with left and right bins always empty

N balls are sequentially and randomly allocated into M bins arranged in a circle. If a bin receives a ball, its left and right bins cannot receive any ball and are always set to be empty (new coming ...
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Easier proof for existence of equilibrium distribution of markov chain

My goal is to get a firm theoretical foundation for the Metropolis-Hastings algorithm and the Gibbs sampler, but none of my classes or books I have found have covered this. I found this paper titled ...
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