Problem $2$: Numerical Recipes

You are given a library function that can generate a non$-$negative Cauchy distributed rv $x$ with pdf

$f_x(x)=\frac{2}{(1+x^2)},x0$

This problem explores "recipes" to synthesize other types of rvs by calling this function and proposing

provably correct post$-$processing steps to realizations of the above rv$.$

In the questions below, you are required to come up with a provably correct procedure to demonstrate that

the desired rv has been generated from $x.$

Derive the cdf of $x$ and its region of support.

Generate a Cauchy rv whose pdf is non$-$zero over $R.$

Generate an $N(0,1)$ rv from $x.$

Generate a $B(n,p)$ rv from $x.$

Generate a $NB(r)$ from $x.$

Generate a rv $Y($and derive its pdf$/$pmf$)$ from $x$ such that:

Its support is the entire negative real axis and the non$-$negative integers.

Conditioned on is Cauchy distributed.

Conditioned on $Y0,Y$ is Poisson distributed.