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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1
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1answer
86 views

Sample uniform direction within cone

My question is pretty much the same as this question below, however I came up with a potential solution to this problem that I didn't see an answer to in the other question and I was wondering if it ...
0
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1answer
36 views

Determining Errors in Monte Carlo Simulation

I was wondering if anyone could throw light on possible errors associated with Monte Carlo sampling. I seem to be getting values that are slightly different each time despite running my model for ...
3
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2answers
120 views

Algorithm to find best in class of groups with weighting?

I have widgets and a single widget will have attributes of: Name Weight (decimal from 0-1) Group (letter A-F) Price (an integer from 1 - 100) I must pick one ...
0
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0answers
23 views

Antithetic pair of non-independent normal random variables

Suppose that I have two non-independent normal random variables, X and Y such that $(X,Y)$ has mean 0 and the following variance covariance matrix: \begin{bmatrix} 1 & \rho ...
0
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0answers
28 views

Understanding graph obtained from Monte-Carlo simulations

I am running a Monte Carlo Simulation where I sample from about 65 Normal Distributions. I also keep track of the probability associated with each sample by approximating a thin area in the Normal ...
0
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0answers
28 views

low discrepancy of halton sequences

I want to proof, that the Halton sequence is low discrepancy. I have to show that $$D_N^*(\mathcal{S})\le C\frac{\ln(N)^s}{N}$$ where $D_N^*$ ist the star discrepancy, $\mathcal{S}$ is the Halton ...
4
votes
2answers
216 views

Monte Carlo estimator of the number of 1's in a very long binary sequence

Preface The question below is related to a problem I am working on, which requires counting the number of times a logic-valued function evaluates "TRUE" given an input value. The size of my input set ...
0
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1answer
123 views

Obtaining useful information from graph obtained via Monte-Carlo Simulations

I've been running Monte Carlo Simulations on some Matlab code and then plot the graph shown below. I was just wondering what useful information I could collect from this graph? Edit: fit ...
1
vote
1answer
36 views

How do we calculate when to shout for the optimal payoff?

For example: A non-dividend paying stock is currently priced at $20, and you hold a put that allows early exercise in 2 months and in 4 months. The option expires in 6 months. Volatility is 30%, and r ...
4
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1answer
98 views

Buffon's experiment with squares

Say, we'd like to make the Buffon's experiment but with squares instead of needles. Notation: $d$ is the distance between lines $b$ is the square side length $y$ is the distance from the center of ...
2
votes
2answers
63 views

How many simulations for a game?

Lately I was interested by Monte-Carlo simulations. I found many papers about this approach in the Internet but for now they are too hard for me. I just want to start understanding this method with ...
0
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0answers
33 views

Average over all positive functions on the unit interval whose Lebesgue integral is one

I want to average over all positive functions on the unit interval whose Lebesgue integral is one. Formally, I want to compute the mean of the following probability distribution defined over function ...
1
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1answer
29 views

Bias Method in monte carlo integration

This is from a proof in my monte carlo course. let $h$ be a smooth function, $T_n = h(\bar{X})$ $\mu = E(X)$ then by taylor expansion $E(T_n -\tau) = E[h(\bar{X} -h(\mu)] = E[\bar{X} - \mu]h'(\mu)+ ...
1
vote
1answer
103 views

Monte carlo estimation of maximum likelihood estimators

I'm interested in numerically finding the maximum likelihood estimator of a parameter $\theta$, as well as the confidence interval of this estimator. First I'll describe the method I've been trying, ...
2
votes
1answer
54 views

Numerical integration of innocent-looking singular integrand

Consider the rather innocent integral: $$I=\int_{0}^{1}a x^{a-1}dx=1,\quad 0<a<1$$ Numerically, this integral converges awfully slowly, and one must use a recursive method to get anywhere near ...
1
vote
1answer
80 views

Monte Carlo estimations of e

I need to estimate $e$ with a monte carlo method. We only learned the crude monte carlo integration, so I can't use any robust monte carlo simulations. I know that $\displaystyle \int\limits_1^x ...
1
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0answers
61 views

Monte Carlo Integration help.

I need to evaluate the following internal: $I=\int^{+\infty}_{-\infty}\dots\int^{+\infty}_{-\infty} f(x_,x_2,x_3\dots x_n) {1\over (\sqrt{2\pi})^N}e^{-{1\over2}x_1^2-{1\over2}x_2^2 \dots ...
4
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0answers
52 views

$\pi$ Monte-Carlo - Probability that O-Lock hit a Spoke?

(Edit: can someone please help me migrate this to physics stack? I think they would be more interested in helping me out with this problem. Thanks.) I have a bicycle with one of those O-locks on it ...
1
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1answer
32 views

finding the optimal decision value for two dependent random events.

I have been struggling with this problem regarding options (bermuda) for some time now. You can exercise this option on two seperate occasions namely at $T_1$ or $T_2$ with a strike price $E$. The ...
1
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0answers
44 views

Density estimation using conditional Monte Carlo simulation

In Stochastic Simulation: Algorithms and Analysis by Glynn and Asmussen on p 146 they provide the following example. Let $f(x)=a/(1+x)^{a+1}$ be the density of a pareto distribution and let $a=3/2$ ...
0
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0answers
47 views

Monte-Carlo calculation of a pdf

I need to evaluate the two first moments of pdf using mcmc methods. I'm already familiar with Metropolis algorithms in simulating statistical physics problems but I never had to approach a conditional ...
2
votes
0answers
41 views

Product Involving Sines

I'm studying the following product: $$p(a,\omega)=\prod_{k=1}^{\infty}a\sin (k\omega\pi),\quad \omega \in \Bbb R,\quad a\in \Bbb R_+.$$ It's easy to see that for $a\in (0,1]$ this product diverges to ...
0
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1answer
86 views

Monte Carlo method error in Bernoulli random variables

Assume I am flipping an unfair coin. Flipping the coin will be heads with probability $p$ and tails with $1-p$. I have no idea what $p$ is (it could even be $.5$!) Let's say I decide to use the Monte ...
0
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0answers
26 views

Weak convergence of discretization scheme with correction

In this article on the Multilevel Monte Carlo method on page 8, http://people.maths.ox.ac.uk/gilesm/files/mcqmc06.pdf, Giles uses a correction term to improve the weak convergence rate of the lookback ...
0
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0answers
20 views

Logit Nomal Prior Distribution

$$\mu \sim N(\mu_0,\sigma_0)$$ $$ X_i \sim LN(\mu,\sigma_x)$$ Does anyone know any method for finding the posterior distribution $P(\mu|X)$ or at least any idea of how to estimate it numerically. I ...
0
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0answers
18 views

Inner product of Gaussians times another multivariate Gaussian derivation help

Assume that: $$p(R|U,V,\alpha) \sim \prod N(R_{ij}|U_i^{T}V_j,\alpha^{-1})$$ $$p(U|\mu_u,\Sigma_u) \sim N(U_{i}|\mu_u,\Sigma_u)$$ $$p(V|\mu_v,\Sigma_v) \sim N(V_{i}|\mu_v,\Sigma_v)$$ Here $R$ is an ...
1
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0answers
46 views

Monte-Carlo tree search convergence proof

I have been doing some reading about Monte-Carlo tree search for games, recently. The Wikipedia article mentions that the algorithm converges to the minimax evaluation for finite zero-sum two-player ...
1
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3answers
125 views

How do I generate $100$ numbers in $[0,1]$ which are more dense at $0$ and $1$?

I just need to generate random numbers in $[0,1]$ which are more dense at the end points. I first thought of generating two sets of numbers from $N(0,1)$ and $N(1,1)$, and then using those. But that ...
0
votes
1answer
60 views

Monte Carlo estimator

I have hopefully a short/simple question regarding monte carlo estimators. The expected value of a function of a random variable can be defined as: $$E[f(x)] = \int_{-\infty}^{\infty} f(x) p(x) dx$$ ...
2
votes
2answers
45 views

Analytic methods vs Monte Carlo (terminology)

What's the correct terminology to say "We can calculate the probability exactly using pure math, as opposed to Monte Carlo simulation"? Analytically sounds like we need Calculus, which we may not ...
1
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1answer
39 views

Question about the Monte Carlo Algortihm

I was reading the Monte Carlo algorithm for finding the area under a curve, say $y=f(x)$. The algorithm considers, $0\le f(x)\le M$ over the closed interval $a\le x\le b$. My question is,that why is ...
0
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0answers
66 views

Monte Carlo Integration

I was reading this document (I will reproduce the equation): ...
0
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1answer
140 views

Alternatives to Monte-Carlo simulation

Imagine I have a model of economy of a region, which consists of several companies, importers and population. Let's assume that all local companies in question produce food and agricultural ...
1
vote
1answer
91 views

Why Gibbs sampling needn't “remixing”

I am generating $\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \dots, \mathbf{x}^{(n)}$ using Gibbs sampling methods. So I want $\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \dots, \mathbf{x}^{(n)} \sim$ some ...
2
votes
2answers
351 views

Please explain Monte Carlo method

Generally I understand the idea of the Monte Carlo method. However, when I read articles about it, there is always shown an example of calculating pi using a square, into which we insert 1/4th of a ...
1
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2answers
48 views

Advantage gained by Blackjack rule variation

There are many tables, charts and simulations for standard Blackjack variations and the % change in house edge that each rule introduces, like this one, for example. I have come across a Blackjack ...
0
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1answer
44 views

Confusion about Monte Carlo integration

I find I can not really understand the Monte Carlo integration, even I use it for many applications, like stochastic ray tracing. Let us take circle-area-calculation for an example, First, we think ...
0
votes
1answer
139 views

Monte Carlo integration, expected value of the sample mean and expected value of f(x)

I am still progressing in my learning of probability and monte carlo method. I understand a basic MC estimator can be written as: $$\bar x = { 1 \over N } \sum_{i=1}^N f(x_i) \approx E[f(x)]$$ I ...
0
votes
1answer
136 views

Markov Chain Monte Carlo in plain English

I barely know what a markov chain is (I had a terrible teacher) and I probably have an idea of what a stationary distribution is... but I don't know how a Monte Carlo method works and I don't know how ...
0
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0answers
91 views

Monte Carlo Simulation

If we have a random variable 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 parameters, ...
0
votes
1answer
49 views

expected value involving probability of inequality random variables.

I have a question, not sure if this can be solved by calculation or Monte Carlo method For random variable G2+G2*min(2/G2-1,G1) where G1, G2 are indenpendt, G1~Lognormal(mu1,cigma1) ...
1
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0answers
51 views

Gibbs / MCMC sampling for sum of parameters - how to improve slow mixing?

Suppose I have a hierarchical Bayesian model, where my observational prediction, $y'$, is calculated as the sum of other parameters, ${\alpha_i}$. My observation equation (the likelihood) is: $P(y | ...
1
vote
1answer
112 views

What is the typical method for sampling uniformly in a convex polytope

The polytope in my case is the intersection of the k-plane $Ax=b$ and $\{x>0\}$ where $A$ is the constraint matrix and $b$ is some solution. I'd like to find a method that is fast and efficient for ...
1
vote
2answers
149 views

Monte Carlo Integration - determining if a random x,y coordinate falls within the circle or square

My textbook says you can take any random (x,y )coordinate between -1 and 1 like (-.3, .5) or (.4, -.7) and determine if the given coordinate falls within the circle if you calculate $\sqrt(x^2+y^2)$ ...
13
votes
3answers
733 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 ...
0
votes
1answer
130 views

Conditional probability of the sum of r.v.

I have $n$ independent random variables $X_i$ with known PDF and CDF (say, Normal, but not necessarily with the same parameters). Given $U_1, U_2 \subseteq \{1,...,n\} $ such that $U_1 \cup U_2 = ...
0
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2answers
42 views

Numerical estimation of simple integral

Considering the problem of numerical evaluation of the integral of a 'good' function $f(x)$ over a unit interval $I = \int_0^1f(x)dx$ Why can we say $I = E[f(U)]$, where $U\sim Uniformly[0, 1]$?
2
votes
2answers
50 views

Maximum of $3x^2e^{-x^3}$

I have a PDF which looks like: $f(x) = 3x^2e^{-x^3}, \quad x \geq 0 $ I need to find it's maximum (to sample from it using the rejection method), so I differentiate and set the result to $0$: ...
0
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0answers
38 views

Monte Carlo Technique and Convergence

I am solving $I = \int_{0}^{1}f(x)dx$ by Monte Carlo, e.g $I = E[f(U)]$ where $U\sim unif([0,1])$ so have $I = E[f(U)]\approx \hat I_M\colon= \frac{1}{M}\sum_{m=1}^Mf(u_m)$ I am interested in how ...
1
vote
1answer
72 views

Bound of the variance of a random Variable

I am having trouble trying to prove that given a random variable $Y$ where $0 \lt m_1 \lt Y \lt m_2 < \infty$, where $m_1$ and $m_2$ are constants the $\displaystyle Var(Y) \le \frac{(m_2 - ...