Monte Carlo Methods to Estimate Area

The area of the ellipse given by + =1 is estimated using three different Monte Carlo procedures.
First, an ordinary hit-or-miss approach, which generates random points in the range , and then estimates the area as the proportion of those points that fall inside the ellipse multiplied by 8
Second, the procedure known as crude Monte Carlo, which looks at the integral giving the area beneath the curve as the expected value of some function, and then estimates it using the mean of that function evaluated at a number of random numbers with the corresponding probability distribution.
Thirdly, a refinement of the latter which uses antithetic random variables, that is, variables that are negatively correlated and which, adequately combined, produce an extremely good estimate.
The behavior of the three approaches may be observed in the graph above. All of them provide an unbiased estimate of the area (equal to 8.89 in the example), but the antithetic variables method is seen to produce a more stable result as the sample size increases, while the other two exhibit a more fluctuating pattern.
Move the slider to see the effect of sample size on the estimation. Press the button to obtain a new set of random samples.



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+