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

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20
votes
2answers
6k views

Probability that a stick randomly broken in two places can form a triangle

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 ...
2
votes
4answers
1k views

Approximating $\pi$ 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 ...
2
votes
2answers
362 views

variance of 26 cards chosen from a deck

Suppose I have a well shuffled deck and I am trying to find the variance of 26 cards randomly chosen without replacement from a deck, assuming the values are from 1 to 13 for the cards. Since the mean ...
6
votes
1answer
3k views

Expected Value and Variance of Monte Carlo Estimate of $\int_{0}^{1}e^{-x}dx$

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 ...
3
votes
2answers
100 views

Mutual information of discrete RVs which converge in distribution to a continuous RV

We have a sequence of pairs of discrete, real-valued RVs $X_n$ and $Y_n$. Each pair is characterized by a discrete probability measure on $\mathbf{R}^2$, which we will just denote $\mu_{X_n,Y_n},$ ...
4
votes
1answer
59 views

Bootstrap method failing where blocking works

I'm computing an average of individual samples that are not entirely independent and need an estimate for the true standard deviation. According to Newman and Barkema's book the most reliable method ...
3
votes
3answers
469 views

Buffon needle experiment for $\pi$ approximation

What are the "best" values for length of needle $(l)$ and distance between paralles $(d)$ for an accurate approximation of $\pi$? Does it have to be $l=d=1.0$ or $l<d$ or $l>d$? Thank you
3
votes
1answer
441 views

Why Monte Carlo integration is not affected by curse of dimensionality?

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)?
2
votes
2answers
102 views

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 $\int_0^1x^2\,dx$. ...
2
votes
0answers
125 views

Evaluating integrals over a domain in $R^n$ using space filling curves

Is there a methodology to simplify evaluation of multi-dimensional integrals using space-filling curves parameterized by a scalar parameter? I am interested in evaluating the integral of a function ...
1
vote
0answers
51 views

Standard deviation of the mean through bootstrap resampling of dependent samples

I'm trying to do a Monte Carlo approximation of an integral where the samples are not independent (how much so can be tuned by a parameter giving how often I sample). Therefore the regular expression ...
1
vote
1answer
53 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
1answer
98 views

Use of this condition on the instrumental density in importance sampling?

From Rubinstein's Simulation Monte Carlo Method, assume r.v. $X$ has density function $f$, $H$ is a measurable function, and $g$ is another density function. If $g$ dominates $Hf$ in the sense that ...
1
vote
2answers
773 views

Monte Carlo simulation on sphere: unbiased random steps

Im doing a Metropolis Monte Carlo simulation with particles on a sphere and have a question concerning the random movement in a given time step. I understand that to obtain a uniform distribution of ...
0
votes
2answers
76 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
39 views

Looking for Method to evaluate the optimal node rate vs number of simulation rate in a Monte Carlo simulation

I am currently working on evaluating an American Option using a Monte Carlo simulation, and I am getting answers but they vary quite a bit. The two variables that I can alter are number of simulations ...
0
votes
1answer
140 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
78 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
200 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 ...