Past exam of the probability, statistics and options I course of the University of Oxford / 2007 / M / 1

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- Solution 1

Solution

Uniform area distribution on the unit disk gives

f_{X, Y} (x, y) = π^{- 1} 1_{{x^{2} + y^{2} \leq 1}} .

The Jacobian determinant has absolute determinant

r

, so

f_{R, Θ} (r, θ) = \frac{r}{π} 1_{{0 < r < 1, 0 \leq θ < 2 π}} .

Integrating out the other coordinate gives

f_{R} (r) = 2 r 1_{(0, 1)} (r), f_{Θ} (θ) = \frac{1}{2 π} 1_{[0, 2 π)} (θ) .

The joint probability density function factors into these two marginal distribution, proving independence of

R

and

Θ

. Finally

E R = \int_{0}^{1} r (2 r) d r = \frac{2}{3} .

The angle at

R = 0

can be defined arbitrarily, since that event has probability zero.

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