# Tagged Questions

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

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### 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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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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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### Mutual information of discrete RVs which converge in distribution to a continuous RV

We have a sequence of pairs of discrete, real-valued RVs $X_n$ and $Y_n$. Each pair is characterized by a discrete probability measure on $\mathbf{R}^2$, which we will just denote $\mu_{X_n,Y_n},$ ...
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### Simulating Random Vectors Based on Conditioning

I'm working on a project where I need to simulate random vectors $(Y, X_1,\dots,X_n)$ in order to understand the joint distribution $f(y,x_1,\dots,x_n)$. I wish to simulate enough random vectors so ...
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### Gibbs sampling truncation for contrastive divergence

I am following Yoshua Bengio's Learning Deep Architectures for AI and at page 31 there is a phrase that confuses me. Starting by lemma 7.1 in the same page: Lemma 7.1. Consider the Gibbs chain ...
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### Lagrange multiplier and minimum variance

Looking into a control variate technique of Monte Carlo simulation I have run into a cost-optimization problem that I'm not quite sure I understand. It seems it has to do with Lagrangian multipliers, ...
Suppose that I am interested in estimating the area of $\Gamma \in \mathbb{R}^2$. I do not know the exact shape of $\Gamma$ but I have a sufficiently large number of sample points $(X,Y) \in \Gamma$ ...