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Box–Muller transformation

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Abstract The generation of pseudo‐random numbers is critical for modern statistical computing. Given any of the well‐tested pseudo‐random generators for the uniform distribution, the probability integral transform may be employed to provide an exact algorithm for transformation to any desired probability distribution. However, if the cumulative distribution function does not afford a simple form, then this strategy is not effective. In particular, the cumulative distribution function of the normal density is not easy to work within this framework. This article describes the ingenious transformation described by Box and Muller in 1958. WIREs Comp Stat 2011 3 177–179 DOI: 10.1002/wics.148 This article is categorized under: Algorithms and Computational Methods > Random Number Generation

The bivariate normal density using polar coordinates (r,θ).

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The densities of a N(0,1) and . The difference is shown times 10.

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A histogram of one million normal variates, each generated using the sum of 12 uniform variates. The true normal density is shown in red.

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Generation of a random variable x given a single uniform random sample u.

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