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

learn more… | top users | synonyms

1
vote
0answers
67 views

Monte Carlo sampling and Kolmogorov–Smirnov test in practice

I have two deterministic algorithms, Algorithm 1 and Algorithm 2. The first has $m$ inputs and one output, and the second has $n$ inputs and one output. The distributions of the inputs of the ...
1
vote
0answers
27 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 ...
1
vote
0answers
49 views

Linear Filtering Problem (Keynman Fac/Particle Model)

$lienar Filtering Problem $$X_n^1 = X_{n-1}^1 + \epsilon_n *W_n $$ $$X_n^2 = (1-\alpha* \delta) X_{n-1}^2 + \beta*\delta X_n^1 $$ $$X_n^3 = X_{n-1}^3 + \delta*X_n^2$$ above is $$\approx$$ $$dX_n^1 ...
1
vote
0answers
35 views

Normalisation of Monte Carlo overlap

Background: In quantum mechanics you are sometimes required to compute the overlap (inner product) of two wave functions (square integrable complex functions) $\Psi(x)$ and $\Phi_i(x)$ as $$ ...
1
vote
1answer
33 views

Find Monte Carlo Variance When Expected Value is not Known

I'm working on a problem that can be approached in two different ways. Both are Monte Carlo algorithms--but it's a hard problem, so I am unsure whether the expected values are indeed the same. I ...
1
vote
0answers
67 views

Convergence of Monte Carlo simulation

I am not sure if this is a valid question but here goes. For the monte carlo method I know that estimation of the mean is also a random quantity and follows a normal distribution. The standard error ...
1
vote
0answers
106 views

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 ...
1
vote
0answers
306 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 ...
1
vote
1answer
109 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 ...
1
vote
1answer
45 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 ...
1
vote
0answers
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 ...
1
vote
1answer
38 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
vote
0answers
61 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 $a=3/2$ ...
1
vote
0answers
54 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
vote
1answer
103 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 ...
1
vote
2answers
51 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 ...
1
vote
0answers
61 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
80 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 - ...
1
vote
1answer
218 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 ...
1
vote
1answer
52 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 ...
1
vote
0answers
113 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 ...
1
vote
0answers
47 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 + ...
1
vote
0answers
156 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 ...
1
vote
0answers
115 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 ...
1
vote
0answers
280 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 ...
1
vote
0answers
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: ...
1
vote
0answers
62 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 ...
1
vote
0answers
167 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), ...
1
vote
0answers
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 ...
0
votes
1answer
346 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 ...
0
votes
2answers
73 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?
0
votes
1answer
58 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 ...
0
votes
2answers
47 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 $U\sim Uniformly[0, 1]$?
0
votes
2answers
189 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: ...
0
votes
2answers
156 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, ...
0
votes
1answer
40 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 ...
0
votes
2answers
504 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 ...
0
votes
1answer
99 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 ...
0
votes
1answer
470 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 ...
0
votes
1answer
178 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 ...
0
votes
1answer
1k 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; ...
0
votes
2answers
29 views

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).
0
votes
1answer
92 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 ...
0
votes
1answer
131 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
votes
1answer
81 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: $$E[f(x)] = \int_{-\infty}^{\infty} f(x) p(x) dx$$ ...
0
votes
1answer
153 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 ...
0
votes
1answer
75 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
176 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
144 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} ...
0
votes
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 ...