![]() ![]() None of this is really surprising mathematically, this is the equivalent of a binomial random variable being divided by its number of trials.īlablabla, who cares about all the math, am I right? Here's a pretty circle, numbers flashing across the screen, and a pretty neat chart plotting our estimated value (powered by SmoothieCharts). Thus as we can see, the expected value of our estimation is π, and the variance of our estimation from the true value decreases as the number of trials increases. Variance of M n, Var = Var⁄ n = ( π⁄ 4)(1 - π⁄ 4)⁄ n Monte Carlo PI Approximating using Monte Carlo Simulation You probably know from your schools math curiculum, that the area of a circle with radius 1 1 is. Let M n be the proportion of points that fall within the circle. Let sample i be represented by X i that takes the value of 1 if point i falls within the circle, and 0 if it doesn't. The more sample points we take, the smaller our deviation from the true value of π. But this general technique, called Monte Carlo Methods, of solving hard deterministic problems with easier probablistic solutions is really powerful and really cool. In this case, since we are carrying out this simulation with JavaScript, floating point errors eventually thwart our efforts at achieving the ultimate prize of super precise approximations of π. Normally to estimate a value of π, we'd have to draw and measure circles, or use outside knowledge of rules of thumb (such as approximating π to 3.14 or 22⁄ 7), or solve an infinite series. We then generate a large number of random points within the square and count how many fall in the enclosed circle. ![]() This methods consists of drawing on a canvas a square with an inner circle. Using this method, we take a hard problem - estimating the value of π - and solve it with a very simple algorithm that you can replicate, without much thinking, by throwing darts blindfolded. One method to estimate the value of (3.141592.) is by using a Monte Carlo method. The area of a circle of radius 1 is π, so the proportion of points that fall within satisfy X 2 + Y 2 ≤ 1 times the area of the square (which is 4) gives us an estimate of the area of the circle and hence the value of π. (points in the circle)/(total points) (piradius2)/(2radius)2. We then count the proportion of generated points that fall within a circle of radius 1 centred at (0, 0). In this Monte Carlo experiment, we generate random X and Y cordinates from -1 to 1. What's going on? Is something going to blow up? ![]()
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