# Computational Efficiency of Polar and Box Muller Method: Using Monte Carlo Application

1884 WordsJul 26, 20128 Pages

CATEGORY: APPLIED MATHEMATICS
Computational Efficiency of Box-Muller and Polar Method Using Monte-Carlo Application
by : Joy V. Lorin-Picar
Mathematics Department
Davao del Norte State College, New Visayas, Panabo City picar_joy@yahoo.com ABSTRACT The efficiency of Mean Square Error (MSE) of the random normal variables generated from both the Marsaglia Polar Method and Box-Muller Method was examined for small and large n with Monte-Carlo application using MATHLAB. The empirical results showed that MSE of the random normal variables using the Marsaglia Polar Method approaches zero as n becomes larger. Moreover, when run in MATHLAB, the Box-Muller method encountered some problems like: a) it runs slow in*…show more content…* The derivation is based on the fact that, in a two-dimensional cartesian system where X and Y coordinates are described by two independent and normally distributed random variables, the random variables for R2 and Θ in the corresponding polar coordinates are also independent and can be expressed as
and
C. Marsaglia Polar Method The Marsaglia Polar Method is a method for generating a pair of independent standard normal random variables by choosing random points (x, y) in the square −1 < x < 1, −1 < y < 1 until
which returns the required pair of normal random variables as
If u is uniformly distributed in the interval 0 ≤ u < 1, then the point (cos(2πu), sin(2πu)) is uniformly distributed on the unit circumference x2 + y2 = 1, and multiplying that point by an independent random variable ρ whose distribution is
will produce a point
whose coordinates are jointly distributed as two independent standard normal random variables. The polar form takes two samples from a different interval, [−1, +1], and maps them to two normally distributed samples without the use of sine or cosine functions.
Two uniformly distributed values, u and v are used to produce the value s = R2, which is likewise uniformly distributed. This polar form is attributed by Devroye[1] to Marsaglia. It is also mentioned without attribution in Carter.[2] Given u and v, independent and uniformly