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

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Continuity of Monte-Carlo simulations with uniformly distributed input parameters

Suppose a continuous and monotone function $f:\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ to be given. So, in the general case, if I slightly change parameters $a$ and $b$, the function ...
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13 views

Cdf of truncated distribution

Let $X$ be a random variable with density $f_x$ and distribution function $F_x$. Define the interval $I = (a,b)$. Given that we know these and the inverse distribution function $F^{-1}_x$, how can we ...
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15 views

Is the result of a Monte-Carlo simulation of a continuous function and with continuous input distributions again continuous?

Is the result of a Monte-Carlo simulation of a continuos function and with continuos input distributions again continuous? Suppose, we have a continuos function $f$ and a number of continuous random ...
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41 views

What is the benefit of stochastic models over deterministic models? [duplicate]

I have posted a similar question earlier and I guess this sounds naive to all of you, but nonetheless let me just ask: Consider I have a simple and deterministic model $M$, with a number of input ...
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1answer
51 views

Monte-Carlo simulation with sampling from uniform distribution

I used to work with Monte-Carlo simulations for a while. In my case, I generated random data for a variety of input parameters according to uniform distributions (with non-negative support), say for ...
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11 views

The exact usage of Sequential Monte Carlo for distributions over time?

I have wondered the usage of Sequential Monte Carlos and it is used as an alternative to Kalman filter for example. However I wonder if this can be also used for simulating a distribution over time? ...
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1answer
38 views

Average of Monte Carlo simulations of continuous functions again continuous?

I hope the following question is clear: Suppose, we have a continuous functions $f:\mathbb{N}^2 \rightarrow \mathbb{N}$. Now, suppose we run Monte Carlo simulations on the function, where the input ...
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28 views

How to estimate a distribution from samples in a histogram

Given a r.v. $\tau$ , I've computed $\Bbb{P}(\tau >a)=e^{-Nx}(e^{Nxe^{-a}}-1) $ , where $N\in\Bbb{N}_{>1} $ and $ x\in \Bbb{R}_{>0} $ are just fixed parameters; say $N = 2 $ and $ x = 1$, ...
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1answer
20 views

Calculating success probability variance using Monte Carlo simulation

For a Monte Carlo simulation where each sample can produce a success value (1) or a failure value (0), what is the variance of the probability for success? Given n samples and r successes, the ...
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1answer
35 views
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45 views

Integrating sine with Monte Carlo / Metropolis algorithm

I'm learning Monte Carlo / Metropolis algorithm, so I made up a simple question and write some code to see if I really understand it. The question is simple: integrating sine over 0 to PI. The ...
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27 views

Biggest rectangle inside a random geometric shape

I'm looking for the most efficient algorithm to find the rectangle with the greatest area inside a random geometric shape. The rectangle can be also rotated of course. I am sure that there exists a ...
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18 views

Opposite of Monte Carlo

In this lecture, at 1:08:35, the lecturer goes from $$\text{argmin}\frac{1}{N}\sum\limits_{i=1}^{N}\text{log}\frac{p(x_i|\theta_0)}{p(x_i|\theta)}$$ to ...
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17 views

Monte Carlo integration and variance

With the monte carlo integration of a function f(x), what do they mean with the variance? Is it the variance of the function we want to integrate? $I = ∫^{\inf}_{inf} f(x)p(x) dx$ (with p(x) some ...
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10 views

Numerical method to fit arbitrary 3D curve by distributing perturbing elements on a 2D grid

I am looking for help in choosing and possibly implementing an appropriate algorithm or method to solve the following problem: I have a surface that has a property $A(r)$ that I want minimized. I ...
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26 views

Intuition behind rejection sampling proof

I have a quick question about the proof of rejection sampling. Suppose we know how to sample from a distribution with $Y$ pdf $q$, and want to sample from a distribution $X$ with (known) pdf $\pi$. ...
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1answer
47 views

Comparing Monte Carlo estmated PI and the real value PI

A famous example of Monte Carlo integration is the Monte Carlo estimate of PI. The unit disk { (x, y) : x2 +y2 <= 1 } is inscribed in the square [ 1, 1] x [ 1, 1], which has area 4. If we ...
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13 views

Monte Carlo Importance Sampling - Finding the new distribution

I'm currently working on a project which requires the implementation of importance sampling to reduce the variance when pricing an option. I think I understand the theory behind what should be going ...
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25 views

Calculating integral with antithetic variables

Use simulation with antithetic variables and find $$\int_{-\infty}^\infty \int_0^\infty \sin(x+y)e^{-x^2+4x-y} \, dx \, dy.$$ so, my question and doubt is how struggle with the infinite limit ? It ...
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1answer
26 views

Mistake in generating random numbers - no irrational ones

Hi I just wondered if the probability densities have to be corrected when using them on a PC since the number representation is not at all continuous. So we cant simulate any irrational numbers and ...
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1answer
56 views

Matlab code, approximate an integral using Monte-Carlo method.

so i have to program the approximation of these two integrals using Monte-Carlo method: $$\int\int_D e^{x^2+y^2} \, dy \, dx $$ $$D=\{(x,y) \in \Bbb R \mid x^2+y^2\le9\}$$ and: $$\int_0^2 ...
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41 views

Ammunition Depot: Monte Carlo Method

I was given the following question from a friend of mine and I can't seem to understand it to well: A squadron of 10 bombers attempts to destroy an ammunition depot. The fighter jet flies in the ...
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17 views

Optimization of Inputs to Monte Carlo Simulation Based on Outputs

I have an optimization process that seems to work, but I want to better understand why it works and whether there's a better way to do what I'm trying to achieve. Basically I am optimizing two (or ...
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15 views

Do Monte Carlo perturbations capture all the uncertainty in prediction?

I have a model $M$ that I use to predict a value $y = M(\vec x)$. I have known one-$\sigma$ error bars on each input $x_i \in \vec x$. I want to know the one-$\sigma$ error bar on my prediction $y$. ...
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1answer
70 views

monte carlo simulation - confidence intervals construction

I am starting with Monte Carlo Simulation. I have run simulation to estimate the mean and the variance of the exponential distribution. Simulation: I have generated random sample from uniform ...
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44 views

Validity of Monte Carlo

My question regards the fundamental validity of the concept of Monte Carlo. In the text where I learned about Monte Carlo some time ago and also on all resources I found on the internet, all authors ...
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1answer
16 views

Efficient methods for drawing random numbers and Monte Carlo for Tsallis q-Gaussians

I would like to draw random numbers from the q-Gaussian used in "Tsallis statistics." This is specifically the distribution $$ f(x) = {\sqrt{\beta} \over C_q} e_q(-\beta x^2) $$ where $$ e_q(x) = ...
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28 views

How to use the Monte Carlo method with curve fitting

I have an assignment in which I need to choose statistical data from the Australian Bureau of Statistics and fit a curve to it. I've chosen the slaughtering of bulls in NSW (first Excel file, ...
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1answer
51 views

How does the Metropolis Algorithm work? (for idiots)

I have the mathematical skills of a house brick and I am desperately trying to learn this algorithm from a computer science perspective. Below is my knowledge of the algorithm. Can someone please ...
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48 views

Integral over space filling curve

Generalizations of the Dirac delta ($\delta$) function ([1]) seemingly enable the expression of $d-1$ dimensional (surface) integrals as $d$ dimensional (volume) integrals in the following form: ...
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1answer
35 views

Equivalence of Probability spaces. Monte carlo integration

Pondering about the independence of dimension of Monte Carlo Integration, I came up with the following explanation: An integral over a square is not harder, thus has the same rate of convergence, ...
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1answer
58 views

How Can ı validate my Monte Carlo Simulation?

Now I am writing my thesis The topic is about error analysis on turbocharger test bench. In order to estimate uncertainties at test bench first of all I used Monte Carlo Method. Now I have to ...
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1answer
29 views

Concentration inequality without variance

Let $X$ be a positive random variable with $\Bbb E X \leq M$. I would like to compute the expectation using Monte-Carlo method, so I am looking for the bounds on $\Bbb P(|\bar X_n - \Bbb E X| \geq ...
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6 views

Generating a sample of Multivariate Epanechnikov Kernel

Is there an efficient algorithm to generate random deviates from a multivariate Epanechnikov kernel? I know it can be performed by rejection sampling but it is ineffective in high dimensions.
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19 views

Parameter Values From Asymmetric Probability Distributions

I am performing a Multipole Decomposition Analysis on some experimental data, essentially fitting a set of experimental data with a linear combination of functions. Annoyingly these functions are not ...
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1answer
15 views

Unbiasedness Importance Sampling

In the 2003 survey paper on MCMC methods by Andrieu et al, there is a section on importance sampling. More specifically, in the section included above it is claimed that $\sum_{i=1}^N ...
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19 views

Metropolis algorithm and formula

In the article by M. Creutz and B. Freedman "Statistical Approach to Quantum Mechanics" authors provide a formula for Metropolis algorithm: $$ W(x_j, x'_j) = \frac{1}{N_0} \left( \theta \left[ S(x_j) ...
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1answer
77 views

Generation of random variable from a complicated CDF

Suppose I am given a CDF of a distribution, given by $F(x) ∝ \int_0^1 x^y e^{-y} dy.$ Here,'x' ranges from 0 to 1. How do I generate a random variable from this distribution?
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42 views

What calculus material can prepare me for MCMC?

I am looking to revise calculus from scratch to move on to Monte Carlo Markov Chain Methods and Quasi Monte Carlo Methods. I studied calculus properly a couple of years ago however it was back in high ...
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36 views

Parametrization of the $Ax=b, x \geq 0$ domain for Monte-Carlo simulation

I have a linear system, $n=15$, with $6$ constraints. There's no problem finding a single solution or establishing the null space; so I can see the full solution space. But I'm only interested in ...
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52 views

Monte Carlo Integration via Ray Casting

Let's suppose we have a 2D line segment $S$ at $y=0$ and extending from $x=-h$ to $x=+h$. We define a function $f(x)=1$ for every point $x$ of $S$. We wish to integrate $f$ over $S$ with Monte Carlo ...
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1answer
94 views

Optimal algorithm for guessing random variable

Let's say you have some unknown quantity $$X\in [0,1]$$ We have N tries to guess the value of X - if you guess $$g_{i}\le X$$ then you capture value $$V_{i} = g_{i}$$ while if your guess is over the ...
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1answer
48 views

How can one read the order of convergence from a loglog-graph?

I am making a task which includes running a Monte Carlo simulation and calculating the order of convergence experimentally. I have to calculate (or approximate) the order of convergence using ...
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1answer
17 views

Inner product estimator - random variable

I'm curently working on the functional space $L^2(\mathbb{R}^n,B(\mathbb{R}^n),\mathbb{P}_X)$ where $\mathbb{P}_X$ is a probability measure. If I generate randomly $N$ realizations of $x_i$ following ...
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1answer
57 views

Approximation to a compounded Binomial distribution

I need to find an approximation, from which I can easily sample, to the following compounded Binomial distribution: $X \sim \mathrm{Binomial}(e^{-\epsilon}, \ n)$ where $\epsilon \sim ...
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34 views

How do I solve a under-determined quadratic multi-variate system?

I have the following equation: $$ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \beta_{11} X_{1}^2 + \beta_{22} X_{2}^2 + \beta_{33} X_{3}^2 + \beta_{12} X_{1} X_{2} + \beta_{23} X_{2} ...
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23 views

How to compute transition probabilities?

I have a stationary process $(X_t)_{t \geq 0}$ with distribution $$\mathbb{P}[X_t \in A ] = \int_A f(x) \, dx$$ for any measurable set $A$ and any $t \geq 0$. I want to compute $$ \mathbb{P}[X_\tau ...
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25 views

Markov-Chain Monte-Carlo: Are transformations on the inputs valid?

The problem: I am trying to solve a high dimensional (up to ~50) class of data fitting & modelling problems. The user specifies the problem, so I would like to make the configuration as easy as ...
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1answer
54 views

Use of ergodic theory in numerical simulations

Is ergodic theory used in numerical simulations? The kind of application I have in mind is: for $\alpha$ irrational, $( n\alpha \mod 1)_{n \geq 0}$ is equi-distributed on $[0,1]$, and I imagine that ...
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11 views

Correct way to implement event rejection in Gillespie method

I am using the Gillespie method for use when generating snowflakes from copy-rotate-translate of basic geometric shapes. The model consists of a matrix, describing coalescence probabilities, and ...