Question

let (Xy) have a vorfarm distribution on the unit circle n. the plane Y are not indeperdert

Answer 1

The joint PDF is given by

$f(x,y)={1\over 2\pi} ~~~~,~~x^2+y^2\leq 1$

a) The marginal PDF of X is

$f(x)=\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}{1\over2 \pi} dy={1\over \pi}\sqrt{1-x^2}$   

and marginal PDF of Y is

$f(y)=\int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}{1\over 2\pi} dx={1\over \pi}\sqrt{1-y^2}$

Clearly $f(x,y)\not=f(x)f(y)$

b) For the region $x^2+y^2<{1\over 4}$

Put $x=r\cos\theta,y=r\sin\theta$ Then $0\leq \theta\leq 2\pi ~,~0\leq r\leq {1\over 2}$

Hence,

$P(X^2+Y^2<{1/4})=\int_{0}^{2\pi}\int_{0}^{1/2}{1\over 2\pi} dr d\theta ={1\over 2}$