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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-36/3/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 36 / 3 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

Generate independent

U, V \sim Unif (0, 1)

and use the Box-Muller transform

G_{1} = - 2 lo g U cos (2 πV), G_{2} = - 2 lo g U sin (2 πV) .

For

R = - 2 lo g U

,

P (R > r) = e^{- r^{2} /2}

, so its density is

r e^{- r^{2} /2}

on

r > 0

. The angle

2 πV

is uniform on

[0, 2 π)

and independent of

R

. Dividing the joint radial-angular density by the polar-coordinate Jacobian

r

gives

f_{G_{1}, G_{2}} (x, y) = \frac{1}{2 π} e^{- (x^{2} + y^{2}) /2} = \frac{e ^{- x^{2} /2}}{2 π} \frac{e ^{- y^{2} /2}}{2 π} .

Thus both outputs are independent and have the standard normal distribution. Independent pairs of uniforms give further independent normal observations. The null event

U = 0

is excluded in implementation so the logarithm is finite.

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