Please solve by PYTHON $($Question$1)$: In the first problem, we would like to demonstrate the law of large number.

i$.$ Generate $n=10$ independent normal random variables $x_1,x_2,$cdots,$x_n$ with mean $1$ and

variance $2($e$.$g$.,$ use the numpy package$).$ Calculate the sample mean widehat$({)}_b=\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i.$

ii$.$ Repeat step i$.$ for $b=1,2,$cdots,$500,$ store those widehat$({)}_b.$ After that, please claculate the sample

$2\mathrm{.}5\%$ and $97\mathrm{.}5\%$ sample quantile of widehat$({)}_b,b=1,2,$cdots,$500.$

iii. Do step $1$ and $2$ for $n=40,90,160,250,360$ and plot the quantiles. The result should LOOK LIKE the following figure: