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Exercise $1$: Monte$-$Carlo integration

The equation of the circle with unit radius centered around $(0,0)$ is given by

$x^2+y^2=1.$

Exploiting the symmetry of the circle, one can define the area under the positive quadrant

to be

${\displaystyle \underset{0}{\overset{1}{}}}ydx={\displaystyle \underset{0}{\overset{1}{}}}\sqrt[\phantom{2}]{1-x^2}dx=\frac{}{4}$

Using the Monte Carlo technique evaluate the integral to find the value of $.$

$($a$)$ Generate $N_{smpl}=10000$ random values for $x,$ distributed uniformly between $0$ and $1.$

Use these values to calculate the average function value $($:$y$:$)$ and estimate $.$

$($b$)$ Repeat the exercise $($a$)N_{\text{e}\text{x}\text{p}\text{t}}=1000$ times to evaluate the uncertainty due to the

limited sample size used in the estimation of $.$

$($c$)$ For different integration samples sizes $($i$.$e$.N_{smpl}={10}^i$;$i=1,$dots,$6)$ which are repeated

$N_{\text{e}\text{x}\text{p}\text{t}}=100$ times, evaluate the mean of the absolute difference between estimated ${}_{\text{e}\text{s}\text{t}}$

and exact result $($np$.$pi$)$ i$.$e$.{}_{}=|{}_{\text{e}\text{s}\text{t}}-|.$ Make a log$-$log plot of ${}_{}$ Vs $N_{smpl}$

and show that the data can be fit to a straight line of slope $-\frac{1}{2}.$