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:

    “It is true that this man was nothing but an elemental force in motion, directed and rendered more effective by extreme cunning and by a relentless tactical clairvoyance .... Hitler was history in its purest form.”
    —Albert Camus (1913–1960)

    “... the history of the race, from infancy through its stages of barbarism, heathenism, civilization, and Christianity, is a process of suffering, as the lower principles of humanity are gradually subjected to the higher.”
    —Catherine E. Beecher (1800–1878)

    “Three million of such stones would be needed before the work was done. Three million stones of an average weight of 5,000 pounds, every stone cut precisely to fit into its destined place in the great pyramid. From the quarries they pulled the stones across the desert to the banks of the Nile. Never in the history of the world had so great a task been performed. Their faith gave them strength, and their joy gave them song.”
    —William Faulkner (1897–1962)