The following technique exploits the Central Limit Theorem to create approximate samples Z from the standard normal

Question:

The following technique exploits the Central Limit Theorem to create approximate samples Z from the standard normal distribution. (An exact method is given in Section A.9.) The mean of a uniformly distributed random variable U on [0, 1],

A.9 Drawing Normal Samples The Box-Muller Algorithm generates exact N(0, 1) samples. Algorithm 32. Box-Muller

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repeat U ~ U (0,1); U = 2U-1; uniform on -1 to 1 V U(0,1); V = 2V-1; a point in the sqr. until W=U2+V Z, Z2

denoted U ∼ U(0, 1), is μ_U = 1/2 and the variance is σ²_U = 1/12. Therefore by the CLT

Z= Ei=1 Ui -n/2 n/12 (1.50)

is approximately N (0, 1). Algorithm 5. Approximate N(0, 1) Samples inputs: n Z=0 for i= 1,..., n U~U (0, 1)

Generate a histogram from this algorithm with n = 12 and compare it with the standard normal density, (1.6) with μ = 0 and σ = 1. Do the same for n = 48, and n = 108.

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