What is the common sense explanation behind that fact that MC integration is free of “curse of dimensionality” in contrast to deterministic integration rules (e.g. trapezoidal rule)?

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 $\int_0^1x^2\,dx$. I set $w(x) = f(x)/g(x) = x^2/2x = x/2$ Then, I solved for $(1/n)\sum_i^nw(x_n) = (1/n)\sum_i^nx/2$ However, I do not know if this is a […]

I want to evaluated the following integral using Monte Carlo method: $$\int_{0}^{1}\int_{0}^{y}x^2y\ dxdy $$ What I tried using Matlab: output=0; a=0; b=@(y) y; c=0; d=1; f=@(x,y) x^2*y; N=50000; for i=1:N y=c+(d-c)*rand(); x=a+(b(y)-a)*rand(); output=output+f(x,y); end output=0.5*output/N; % 0.5 because it’s the area of the region of integration fprintf(‘Value : %9.8f\n’,output); However this code didn’t give me […]

I’m working on a problem of estimating a function $y=f(x): \mathbb{R}^d \rightarrow \mathbb{R}$. Namely, I have an unknown function $f(x)$ (like a black box), what I can do is to input $x^{(i)}$ to it, and obtain $y^{(i)}$ ($i=1,2,\cdots, N$). Then I get a dataset $(x^{(i)}, y^{(i)})$ and am able to fit a function on it. […]

Given the integral: $I=\int_{0}^{1}e^{-x}dx$, use standard Monte Carlo with 1000 random numbers and repeat the simulation 1000 times. (a) What is the expected value and variance of the simple Monte Carlo estimate of I ? I wrote the following to calculate the Monte-Carlo approximation to the integral: a=0; b=1; n=1000; x=a+(b-a)*rand(n,1); q=1/(b-a); f=exp(-x); i=mean(f./q) How […]

For example, consider a 6 sided die rolled 10 times. Based on the following monte-carlo simulation, I get that the side that appears most will appear 3.44 times on average. n = 6 k = 10 samples = 10000 results = [] for _ in range(samples): counts = {s:0 for s in range(n)} for _ […]

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 yield a more precise result than distributing these points evenly on the domain. More precisely: Let $f:[0,1]^d \to \mathbb{R}$ be a continuous bounded integrable […]

I used to work with Monte-Carlo simulations for a while. In my case, I generated random data for a variety of input parameters according to uniform distributions (with non-negative support), say for example two variables $a$ and $b$ where data is generated: $a_1, \dots, a_n \leftarrow U(1,2)$ and $b_1, \dots, b_n \leftarrow U(4,5)$. Eventually, I […]

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 $0$ to $1$, call the breaking points $x$ and $y$, consider the unit square of the coordinate plane, shade the areas that […]

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..

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