Ratio of uniforms is Cauchy

[JennySton]

Hi all!

I have U is uniform(0,1) and V is uniform(-1,1) such that U^2 + V^2 <= 1. Furthermore, let X=V/U. Show that X is Cauchy.

I tried to solve this problem using transformation:
X = V/U => V = XY
Y = U => U = Y
Then the Jacobian is |J|=y and the joint pdf of X and Y is just f(x,y)=.5*y. This seems to be wrong since integrated out y won't give me a Cauchy.

Then I tried using the CDF:
P(X <= x) = P(V/U <= x) = P(V <= Ux) = x -.5 but this does not work either.

Obviously I have to incorporate the condition that U^2 + V^2 <= 1 but I have no idea how and I would really appreciate some help.

Thanks,
Jenny

[fed1]

Dont we need to know what is joint pdf of <X,Y> ?

Are they independant?

[fed1]

I mean joint pdf of <U, V>. Without this problem is not solvable.

[Martingale]

Hi all!

I have U is uniform(0,1) and V is uniform(-1,1) such that U^2 + V^2 <= 1. Furthermore, let X=V/U. Show that X is Cauchy.

I tried to solve this problem using transformation:
X = V/U => V = XY
Y = U => U = Y
Then the Jacobian is |J|=y and the joint pdf of X and Y is just f(x,y)=.5*y. This seems to be wrong since integrated out y won't give me a Cauchy.

Then I tried using the CDF:
P(X <= x) = P(V/U <= x) = P(V <= Ux) = x -.5 but this does not work either.

Obviously I have to incorporate the condition that U^2 + V^2 <= 1 but I have no idea how and I would really appreciate some help.

Thanks,
Jenny

1) I'm betting the joint pdf is ; one over the area of the circle that U,V are defined on.

2) you then want



3) then compute

[JennySton]

Thanks for your help! I see what I did wrong!