 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2009/m/3/b/i/solution

X, Y

(2 π)^{- 1} e^{- (x^{2} + y^{2}) /2}

. The polar-coordinate Jacobian is

r

, so

f_{R, Θ} (r, θ) = \frac{r}{2 π} e^{- r^{2} /2}, r > 0, 0 \leq θ < 2 π .

Integration over the angular coordinate gives

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

for

r \geq 0

and zero otherwise. Thus

R

Θ

is uniform and independent of

R

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