Hi,

I am trying to draw F (CDF of Cauchy) using X/Y relation for X,Y~N(0,1) iids and at some point very shortly I get lost.

So.

Problem:

(And I know it is Cauchy's disterbution.)

What I try do:

I want to find F_Z whilest Z=X/Y and derive it, whilest I want to find it using explicit drawing of definition areas on X,Y axis. Like that:

This is the area below Y=X/z for X,Y,z>0 on X,Y axis graph. (May I not destroy the text with ugly picture?)

Picture is below.

Now, when z>0 it's all summerizes beautifully and if z would be only >0. Meaning by that, if F_Z(z) would be defined only on the right side of Y axis, we would finish this and I would even do the calculations.

BUT

z (small z) might be <0.

My question is -

I am trying to draw F (CDF of Cauchy) using X/Y relation for X,Y~N(0,1) iids and at some point very shortly I get lost.

**Especially**because I don't know if I'm on the right direction.**Could you direct me to the right way of solving this using the way I chose?**So.

Problem:

**Given X,Y~N(0,1) iids find PDF of X/Y**.(And I know it is Cauchy's disterbution.)

What I try do:

I want to find F_Z whilest Z=X/Y and derive it, whilest I want to find it using explicit drawing of definition areas on X,Y axis. Like that:

**For X,Y,z>0 we get****F_Z(z) = P(z<Z) = P(z<X/Y) = P (Y<X/z)**This is the area below Y=X/z for X,Y,z>0 on X,Y axis graph. (May I not destroy the text with ugly picture?)

Picture is below.

*I think we can use Proposition formula to find how much is it exactly. (?)*

We can use double integrals here.We can use double integrals here.

Now, when z>0 it's all summerizes beautifully and if z would be only >0. Meaning by that, if F_Z(z) would be defined only on the right side of Y axis, we would finish this and I would even do the calculations.

BUT

z (small z) might be <0.

**So for example, for X,Y>0 and z<0, we get****F_Z(z) = P(z<Z) = P(z<X/Y) = 1.**My question is -

**How to apply the complete disterbution function while taking in mind small z**? P(X>0)=P(Y>0)=P(X<0)=P(Y<0)=1/2**But what about P(Z<0)?****Or how do I actually calculate it when for F_Z(z), z<0 and F_Z(z), z>0 the formulas are different?**
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