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The Marsaglia polar method [1] is a pseudo-random number sampling method for generating a pair of independent standard normal random variables. [2]Standard normal random variables are frequently used in computer science, computational statistics, and in particular, in applications of the Monte Carlo method.
The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution. This is a special case when μ = 0 {\textstyle \mu =0} and σ 2 = 1 {\textstyle \sigma ^{2}=1} , and it is described by this probability density function (or density): φ ( z ) = e − z 2 2 2 π . {\displaystyle \varphi (z ...
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It plays 200000 games of craps, finds the number of wins and the number of throws necessary to end each game. The number of wins should be (very close to) a normal with mean 200000p and variance 200000p(1 − p), with p = 244 / 495. Throws necessary to complete the game can vary from 1 to infinity, but counts for all > 21 are lumped with 21.
A modification of Lagged-Fibonacci generators. A SWB generator is the basis for the RANLUX generator, [19] widely used e.g. for particle physics simulations. Maximally periodic reciprocals: 1992 R. A. J. Matthews [20] A method with roots in number theory, although never used in practical applications. KISS: 1993 G. Marsaglia [21]
For an exponential distribution, the tail looks just like the body of the distribution. One way is to fall back to the most elementary algorithm E = −ln(U 1) and let x = x 1 − ln(U 1). Another is to call the ziggurat algorithm recursively and add x 1 to the result. For a normal distribution, Marsaglia suggests a compact algorithm:
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It discards 1 − π /4 ≈ 21.46% of the total input uniformly distributed random number pairs generated, i.e. discards 4/ π − 1 ≈ 27.32% uniformly distributed random number pairs per Gaussian random number pair generated, requiring 4/ π ≈ 1.2732 input random numbers per output random number.