Please write in python.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$,$ cumsum

from scipy.stats import probplot

import pandas as pd

import matplotlib.pyplot as plt

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

$UU$ nif $[0,1],F^{-1}(U)$ 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$)$

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

$($a$)$ Generate $10000$ samples from Bin$(10,0\mathrm{.}4)$ using $($i$)$ the inversion method directly and $($ii$)$ using the inversion method to draw corresponding Bernoulli

distributed samples. $($iii$)$ Plot the resulting empirical distribution functions and add the theoretical distribution function in one figure.