You are implementing a Monte Carlo simulator for network-packet jitter. The only random-number source available returns independent samples from the continuous uniform distribution on (0, 1). To feed the noise model, the simulator must generate pairs of independent standard normal (mean 0, variance 1) random variables on every call. Which one of the following transformations of two independent Uniform(0, 1) samples U1 and U2 will correctly produce the required standard normal variables Z1 and Z2?
The Box-Muller transform maps two independent Uniform(0, 1) variables to two independent N(0, 1) variables by converting the uniform samples into polar coordinates. The correct transformation is:
Z1 = √(-2 ln U1) cos(2π U2)
Z2 = √(-2 ln U1) sin(2π U2)
This works by setting the squared radius R² = -2 ln U1 and the angle Θ = 2π U2. Because the Jacobian of this transformation exactly cancels the standard bivariate normal density, the resulting pair has the desired distribution. The other choices break one or more of the requirements:
Transformations that use π instead of 2π for the angle or take logarithms of both U1 and U2 distort the output, so it is no longer standard normal.
Mixing the two uniform variables inside the logarithm, such as in the expression √(-2 ln (U1 / U2)), changes the radial distribution and makes Z1 and Z2 dependent, so they are neither independent nor standard normal.
Therefore, the transformation using √(-2 ln U1) for the radius and the trigonometric functions of 2π U2 for the angle is the only one that satisfies the specification.
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