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

Table of contents
- i b
  - Solution i
- ii b
  - Solution ii

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

.

ii

Solution

For

w \geq 0

,

P (W \leq w) = P (R \leq w) = 1 - e^{- w /2}

. Differentiation gives

f_{W} (w) = \frac{1}{2} e^{- w /2} 1_{{w \geq 0}} .

This is an exponential distribution of rate

1/2

, with mean two and variance four.

 Ancestors (12)

 View article source

 Discussion (0) New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version