Marsaglia Polar Method - History

History

This idea dates back to Laplace, whom Gauss credits with finding the above

by taking the square root of

I^2 = \int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(x^2+y^2)/2}\,dx\,dy =\int_0^{2\pi}\int_0^\infty re^{-r^2/2} \, dr \, d\theta.

The transformation to polar coordinates makes evident that θ is uniformly distributed (constant density) from 0 to 2π, and that the radial distance r has density

(Note that r2 has the appropriate chi square distribution.)

This method of producing a pair of independent standard normal variates by radially projecting a random point on the unit circumference to a distance given by the square root of a chi-square-2 variate is called the polar method for generating a pair of normal random variables,

Read more about this topic:  Marsaglia Polar Method

Famous quotes containing the word history:

    “A poet’s object is not to tell what actually happened but what could or would happen either probably or inevitably.... For this reason poetry is something more scientific and serious than history, because poetry tends to give general truths while history gives particular facts.”
    —Aristotle (384–323 B.C.)

    “The history of the genesis or the old mythology repeats itself in the experience of every child. He too is a demon or god thrown into a particular chaos, where he strives ever to lead things from disorder into order.”
    —Ralph Waldo Emerson (1803–1882)

    “The greatest horrors in the history of mankind are not due to the ambition of the Napoleons or the vengeance of the Agamemnons, but to the doctrinaire philosophers. The theories of the sentimentalist Rousseau inspired the integrity of the passionless Robespierre. The cold-blooded calculations of Karl Marx led to the judicial and business-like operations of the Cheka.”
    —Aleister Crowley (1875–1947)