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:

    “In history the great moment is, when the savage is just ceasing to be a savage, with all his hairy Pelasgic strength directed on his opening sense of beauty;—and you have Pericles and Phidias,—and not yet passed over into the Corinthian civility. Everything good in nature and in the world is in that moment of transition, when the swarthy juices still flow plentifully from nature, but their astrigency or acridity is got out by ethics and humanity.”
    —Ralph Waldo Emerson (1803–1882)

    “Tell me of the height of the mountains of the moon, or of the diameter of space, and I may believe you, but of the secret history of the Almighty, and I shall pronounce thee mad.”
    —Henry David Thoreau (1817–1862)

    “Systematic philosophical and practical anti-intellectualism such as we are witnessing appears to be something truly novel in the history of human culture.”
    —Johan Huizinga (1872–1945)