Past exam of the probability, statistics and options I course of the University of Oxford / 2009 / M / 3 / b / i

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- Solution i

Solution

The joint density of

X, Y

is

(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

has the Rayleigh distribution, while

Θ

is uniform and independent of

R

.

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