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

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8
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1answer
4k views

Expected Value and Variance of Monte Carlo Estimate of $\int_{0}^{1}e^{-x}dx$

Given the integral: $I=\int_{0}^{1}e^{-x}dx$, use standard Monte Carlo with 1000 random numbers and repeat the simulation 1000 times. (a) What is the expected value and variance of the simple Monte ...
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0answers
13 views

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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0answers
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 ...
0
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0answers
16 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 ...
2
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0answers
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 ...
0
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1answer
53 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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0answers
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? ...
0
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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 ...
1
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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) = ...
0
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0answers
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$, ...
3
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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 ...
1
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0answers
125 views

Monte Carlo Simulation

If we have a random variable $H$, such that $$ H=B_1G_2\min(1,G_1)+B_2\frac{\min(2,G_1 G_2+G_2)}{n-B_1-B_2}, $$ where $n$ is constant, $G_1$ and $G_2$ are independent lognormal with different ...
1
vote
1answer
40 views

Is there a such thing as a quasi-random shuffle?

I've recently experimented with Quasi-random numbers in monte-carlo applications. Is there a way to construct a quasi-random shuffle? By that I mean can I take a sequence $Q$ and shuffle it to produce ...
2
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0answers
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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0answers
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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3answers
576 views

Random directions on hemisphere oriented by an arbitrary vector

Hy, i'm writing a raytracer, and for that I need to generate n random vectors that are inside an hemisphere oriented by the surface normal. Ideally, I would also like being able to restrict the rays ...
0
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0answers
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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0answers
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 $\pi$. ...
2
votes
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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0answers
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 ...
1
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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 ...
2
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0answers
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 ...
1
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1answer
42 views

Multi-armed bandit with infinitely-many arms

Has anyone studied variants of the multi-armed bandit algorithm with infinitely many arms? I have a collection of distributions parametrized by an integer $n$. Unfortunately, I can't analytically ...
0
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1answer
58 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 ...
2
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0answers
42 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 ...
3
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1answer
49 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 ...
0
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0answers
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 ...
0
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0answers
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 $y$. ...
0
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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 ...
2
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0answers
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 ...
1
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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, ...
1
vote
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 ...
22
votes
2answers
6k views

Probability that a stick randomly broken in two places can form a triangle

Randomly break a stick (or a piece of dry spaghetti, etc.) in two places, forming three pieces. The probability that these three pieces can form a triangle is $\frac14$ (coordinatize the stick form ...
0
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0answers
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, ...
0
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0answers
49 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: ...
18
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4answers
2k views

Why does Monte-Carlo integration work better than naive numerical integration in high dimensions?

Can anyone explain simply why Monte-Carlo works better than naive Riemann integration in high dimensions? I do not understand how chosing randomly the points on which you evaluate the function can ...
2
votes
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 ...
2
votes
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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0answers
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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0answers
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 ...
0
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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 ...
3
votes
0answers
48 views

Estimate the volume of a convex body given a uniform random sample of points inside it?

Let $K$ be a convex, full-dimensional, bounded region of $\mathbb{R}^n$. More precisely, there exist two balls of radiuses $0<r<R$ such that the ball of radius $r$ is fully contained inside $K$ ...
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0answers
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) ...
0
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1answer
78 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?
3
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4answers
402 views

What are some uses for Monte Carlo simulations in mathematics?

I've recently been interested in Monte Carlo simulations and their uses, unfortunately most of the examples I find are difficult to understand for a beginner. What are some simple examples of using ...
2
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0answers
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 ...
2
votes
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 ...