In this exercise, we will generate $($pseudo$-)$random numbers using the inversion and accept$-$reject method. In order to generate the random numbers you are

only allowed draw from the Uniform distribution and use

from random import uniform

from scipy,special import binom

from numpy import sqrt$,$ pi$,$ exp, tan$,$ cunsum

from scipy.stats inport probplot

import pandas as pd

Import natplotlib.pyplot as plt

Inversion method: Let $F$ be a distribution function from which we want to draw. Define the quantive function $F^{-1}(u)=$inf$\{x$:$F(x)u,0u1\}.$ Then, if

$U{U}_{ni}f[0,1],F^{-1}(U)$ has distribution function $F.$

Accept$-$reject: Let $f$ be a density function from which we want to draw and there exists a density $g$ from which we can draw $($e$.$g$.,$ via the inversion method$)$

and for which there exists a constant $c$ such that $f(x)cg(x)$ for all $x.$ The following algorithm generates a random variable $x$ with density function $f.$

Generate a random variable $x$ from density $g$

Generate a random variable $UUnif[0,1]($independent from $x)$

If Ucg$(x)f(x),$ return $x,$ otherwise repeat $1,-3.$

The number of iterations needed to successfully generate $x$ is itself a random variable, which is geometrically distributed with the success $($acceptance$)$

probabinty $p=P(\frac{U}{c}g(x)f(x)).$ Hence, the expected number of iterations is $\frac{1}{p}.$ Some calculations show that $p=\frac{1}{c}$

$($b$)$ Generate $10000$ samples from the standard normal distribution using the accept$-$reject method with candidate density $g(x)=((1+x^2){)}^{-1}$ with

distribution funciton $G(x)=ta{n}^{-1}\frac{x}{}$ from the standard Cauchy distribution. To this end, $($i$)$ determine $($mathematically or via simulation$)$ the value of $c1$

closest to one so that $f(x)cg(x)$ for all $x.($ii$)$ Obtain $10000$ standard normal random variables using the accept$-$reject method, generating Cauchy

distributed random variables using inversion method. $($iii$)$ Compare estimated and theoretical acceptance probabilities. $($iv$)$ Generate a QQ$-$plot of the

generated sample.