Montecarlo estimate of a integrand from 0 to ∞\infty

I have a question about monte carlo estimation of integrals. Suppose I am told to estimate using monte carlo, the integral: f(y)=∫y041+x2dx I want to estimate f(∞). I know that with some calculation, the exact values are given by π and 2π. However, there is some confusion with respect to defining the bounds and area … Read more

Error in calculation of π\pi using Monte Carlo method.

So lets say we are trying to calculate value of π using MonteCarlo method. By picking random points in a square and measuring their distance from the center and if k points lie inside the circle using the ratio kN to calculate the value of π. How do I derive the formula for variance of … Read more

Simple Monte Carlo Integration

I am trying to use Monte Carlo Integration, which is nicely described in the answer here (Confusion about Monte Carlo integration). I am using Monte Carlo Integration to evaluate ∫10x2dx. I set w(x)=f(x)/g(x)=x2/2x=x/2 Then, I solved for (1/n)∑niw(xn)=(1/n)∑nix/2 However, I do not know if this is a good solution. I would like to know of … Read more

Approximating \pi\pi using Monte Carlo integration

I need to estimate \pi using the following integration: \int_{0}^{1} \!\sqrt{1-x^2} \ dx using monte carlo Any help would be greatly appreciated, please note that I’m a student trying to learn this stuff so if you can please please be more indulging and try to explain in depth.. Answer Generate a sequence U_1,U_2,\ldots of independent … Read more

Does this count as a Monte Carlo simulation?

Let’s say I have a group of robots that walk on a 11×11 grid of tiles in four directions, N, S, E, W, and each robot has different probability distribution functions that assign different probabilities to each of the four outcomes per move. If I want to know how many moves it will take on … Read more

How can I prove that the error of the Monte Carlo method for finding π\pi decreases as NN, the number of samples, increases?

I am currently playing around with the Monte Carlo method of finding π. The idea is pretty simple, I work with a unit square of length one on each side, with the coordinates of (0,0),(1,0),(0,1),(1,1). Then, I draw on a quarter-circle of radius 1 onto the unit square. It looks something like: Now all there … Read more

Question about Random Walks and An O∗(n5)O^*(n^5) Volume Algorithm for Convex Bodies – Kannan Lovasz Simonovits 97

I’ve been trying to understand this paper: “Random Walks and An O∗(n5) Volume Algorithm for Convex Bodies“, Ravi Kannan, Laszlo Lovasz, Miklos Simonovits. Motivation: The paper is about estimating the volume of convex bodies using a random walk. We need a particular inequality to show that the random walk distributes well inside the convex body. … Read more

Monte Carlo with non uniform weighting

So, I just want to check if what is in my mind is in fact true. Assume, that we have are given a distribution pz(k) over the whole Z+. We are interested in approximating pv(v) over some observed variable v. However, we are not given pv directly, but rather a joint p(v,h)=p(h)p(v|h) such that the … Read more

How to estimate an expectation under expontial random measure by Importance sampling ? Thanks

Say I have to estimate an expectation of a function X^{0.9} under exponential random measure(with density f), i.e., E_f(X^{0.9})= \int_0^ \infty x^{0.9} e^{-x} dx by importance sampling. The exponential tilted measure could be easily derived as f_{\theta }(x)={(1-\theta)e^{({1-\theta})x}} . The optimal tilting parameter would be found by solute the equation \nabla E_{f_{ \theta }} [(X^{0.9}e^{-\theta … Read more

Ensemble approach to simulated annealing

I am reading the paper Ensemble approach to simulated annealing, which contains the following algorithm for adjusting the temperature during annealing: where T is the temperature, \langle E \rangle_T is the equilibrium average energy at T, \langle E \rangle is the average energy of the ensemble, and \langle (\Delta E)^2 \rangle is the variance in … Read more