Please provide code where neccesary

A random variable $x$ has the pmf given below.

$p_x(x)=p(x=x)=\{\begin{array}{cc}0\mathrm{.}1,& x=1\\ \mathrm{.}3,& x=2\\ \mathrm{.}4,& x=3\\ \mathrm{.}2,& x=4\end{array}$

$($a$)$ Calculate the $E[x]$ and var$(x)$ by hand.

$($b$)$ Calculate the $E[Y]$ and var$(Y)$ where $Y=x^2+2x+1$ by hand.

$($c$)$ Now we want to compute the empirical values of all the quantities above by

running a simulation using Python.

i$.$ Write the python code to generate $n$ samples of the random variables $x$ and

$Y$ and calculate the empirical mean and variamce of these samples.

ii$.$ Run you code for $n={10}^k$ for $k=3,4,5,6.$ Print out the results and compare

the results with the statistical values.