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

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What does “sequence is equidistributed in [0, 2]” mean?

I was reading an article in which they are mentioning this sentence: "sequence is equidistributed in [0, 2]" where the sequence in question, is a sequence of real number (the article in question is ...
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284 views

Monte Carlo standard deviation of the mean estimate too small.

I'm doing a Monte Carlo calculation and use the standard deviation of the mean $\sigma_M$ as the error. To get an estimate of this from the regular standard deviation I use $$\sigma_M=\dfrac\sigma ...
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1answer
98 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 ...
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1answer
43 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 ...
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70 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 ...
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37 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 ...
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58 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 $a=3/2$ ...
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51 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 ...
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101 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 ...
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50 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 ...
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59 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 | ...
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77 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 - ...
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213 views

Will I have learned the prerequisites for self learning stochastic calculus and monte carlo method?

I'm an undergraduate econ major, and my main focus is in actuarial sciences, which as you may or may not know it's pretty mathematical. Some of the topics I will have to learn at some point on my own ...
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51 views

variance reduction

Say i have $n$ variables with variances $V_1,V_2,...V_n$. The sum of the variables will have a variance of $V=V_1+V_2+..V_n$ .Now if i am given N total simulations to reduce the variance V, how do i ...
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101 views

Monte-Carlo for SDE with square root diffusion term

I've recently got a question from a Master student about a numerical simulation/integration of the SDE of the following shape $$ \mathrm dX_i(t) = \left(\sum_{j=1}^M \nu_{ji} a_jX_j(t)\right)\mathrm ...
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46 views

Simulating of GBM

I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem: Given a GBM of the form $dS(t) = \mu S(t) dt + ...
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152 views

Volume with MonteCarlo

I'm asked to find the volume of a given bounded solid in $\mathbb{R}^3$ by MonteCarlo means: since it is contained in a prism, I generate random points in the prism and see what proportion of them lie ...
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112 views

Approximation of SDE

I have been struggling with the following problem: If you want to find a numerical result by simulating the paths of a stochastic differential equation, in particular a geometric brownian motion I ...
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260 views

Computing Margin-of-Error using Monte Carlo simulations

I am interested in computing the margin-of-error for a metric computed on a random sample. The underlying distribution (finite) from which the random sampling is done is not normal (its extremely ...
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46 views

How to sample the walk which visits each vertex of a graph specific number of times?

Is there any MCMC mathod that allow me to uniformly sample from all feasible walks where the following restrictions apply: ...
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61 views

Gibbs Sampling versus General Cases

Suppose we are given a prior distribution about an unknown parameter $\pi(\theta)$. Also we are given $f(x_{1}, \dots, x_n|\theta)$. We want to find $\pi(\theta|x_1, \dots, x_n)$. Now ...
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163 views

Markov Chains - Using Gibbs & Metropolis algorithm.

Suppose $f_x,_y$ is bivariate normal distribution. I was given the parameters $(μ_1, μ_2, σ_1^2, σ_2^2)$ and $ρ=0.95$ the correlation coefficient. I want to generate $(x_1,y_1), ...
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42 views

Integrating with respect to an MCMC Kernel

I'm currently trying to evaluate an integral with respect to an MCMC kernel, and I wanted someone else to read what I `proved' to see if it's really correct. It just doesn't feel right. Suppose I ...
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336 views

Why are random numbers necessary for a Monte Carlo simulation?

This may be somewhat of a question with an obvious answer, but I can not seem to understand the necessity of "truly" random numbers to make a Monte Carlo simulation a good one. I understand that not ...
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65 views

Estimate gamma function using monte carlo

Let $\Gamma(\beta) = \int_0^\infty x^{\beta - 1} e^{-x} dx$ how to estimate the above gamma function using monte carlo? Any idea?
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44 views

Random sampling using Metropolis vs. Accept-Reject

I'm working on a project comparing the accept-reject (von Neumann) and Metropolis methods of sampling. I'm generating a sample of size $N$ from a normal distribution $N(1,(\frac{1}{2})^2)$. What I ...
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46 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 $U\sim Uniformly[0, 1]$?
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53 views

Monte Carlo gamma function

This question was asked before but I'd like to ask something more precise given the answer that was given. [ Estimate gamma function using monte carlo ] What is the criterion for a random point to ...
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136 views

How to use Monte Carlo method to find volume?

So the question asks to use the Monte Carlo method to find the volume of an irregular figure defined as: ...
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139 views

Are irreducible, positiv-definite Markov chains aperiodic?

If $M$ is the transition matrix of a discrete Markov chain, and $M$ is both irreducible, symmetric and positiv-definite, is the resulting Markov chain necessarily aperiodic? In my intuition, ...
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38 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 ...
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484 views

How to recognize ellipse/ellipsoid from random points? UN-weighted average?

Suppose we are getting random points in 2D (or 3D) which tend to be on ellipse (or ellipsoid). We can't guarantee points are uniformly distributed over ellipse (ellipsoid surface). The task is to ...
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1answer
98 views

Logic Question Regarding Sample Number and Time Increase

So this question is one of those "I'd rather ask and look stupid now than never know" types of questions. It goes as follows: The error in a Monte Carlo estimate is dominated by ...
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456 views

Combining two triangular distributions to yield one distribution

I am interested in using some Monte Carlo methods to help with an estimation problem I have. I need to allow multiple estimators to estimate line-items giving a best, average and worst case estimate ...
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167 views

Double Integrals & Expected Value Monte Carlo Method

Tell me if I'm wrong Let $\Omega = [a,b]\times[c,d]\subseteq\mathbb{R}^2$, then $$ \iint_\Omega ...
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1answer
841 views

Evaluating double integrals using monte carlo methods in matlab.

I used the monte carlo method to integrate $\int_{0}^{1}x^2dx$ in matlab. My matlab code was simply the following: A=1; N=10000; s=0; for i=1:N x=rand; y=rand; if y<= x^2; ...
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Let Y be a random variable with $0\le Y\le 1.$ [duplicate]

Let Y be a random variable with $$0\le Y\le 1.$$Show that $$var(Y)\le 1/4 $$ and that $$var(Y)= 1/4 $$ if and only if P(0)=1/2=P(1).
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89 views

How to mathematically prove that we are sampling from same distributions?

The content of this question is about rigorously proving something which is otherwise considered easily correct intuitively. Let's assume we have a multivariate distribution $g(x_1,x_2,...,x_n)$ over ...
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123 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 ...
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77 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: $$E[f(x)] = \int_{-\infty}^{\infty} f(x) p(x) dx$$ ...
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151 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 ...
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66 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 ...
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165 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 ...
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1answer
136 views

Monte Carlo sampling a binomial expansion

I want to figure out the following question $$ 1 = (10 - 9)^{100} = 10^{100}-100 \times 10^{99} \ 9 + \frac{100 \times 99}{2} 10^{98} \ 9^{2} - \frac{100 \times 99 \times 98 }{3}10^{97} 9^{3} ...
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1answer
37 views

Why cannot the Markov Chains used in MCMC simulations be null recurrent?

I am aware this question borderlines retardedness, but I am seeking an accurate explanation. I understand in null-recurrent cases, the expected amount of time to explore states can be infinite. Is ...
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467 views

How to decide what is the probability distribution in a Monte-Carlo simulation?

For a Monte-Carlo integration of $$\int_\Omega P(x)f(x)\ \text d x,$$ there seems to be no apriori distinction if $f$ or $P$ is the probability function. So does it matter if I consider $$P, f, P ...
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108 views

Finding a distribution of a random variable generated using a Monte Carlo method

I would greatly appreciate if somebody could confirm or negate my result to the following problem. I am especially not sure about "putting it all together" step. Generate $U_1,U_2,U_3 \sim ...
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Forecasting disputed transaction frequencies

Problem I would like to forecast credit card chargeback/dispute frequencies using historical dispute data I have recorded over time. The data I currently store includes: Disputed transaction date ...
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Estimating the Square of a Mean

Suppose I want to estimate $\theta = (\mathbb{E}[f(X)])^2$, where $f: \mathbb{R} \to \mathbb{R}$ and is Borel-measureable, and $X$ is a random variable. I'll use Monte Carlo, for which one ...
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26 views

Showing that inter-arrivals of a Poisson process are i.i.d with exponential law

Let $(N_t)_{t \in \mathbb{R}_+}$ be a Poisson process and let $(S_n)_{n \geq 1}$ the interarrivals times.For $\varepsilon > 0$ we define $$Y_k^\varepsilon := 1_{\{N_{k \varepsilon} > ...