Exercise $3$: $(6$ points$).$ Let the random variable X follow a distribution with density e$\backslash$kappa cos$($x$)$ f$($x$)=2\backslash$pi I$0(\backslash$kappa $),$ if x in $[\backslash$pi $,\backslash$pi $]0,,$ otherwise $(0\mathrm{.}1)$ where $\backslash$kappa $>0$ is a parameter and I$0()$ is the modified Bessel function of the first kind of order $0.$ For this exercise, you will need to use the rejection sampling algorithm with a proposal that you deem appropriate. $($a$)$ Derive mathematically the expected number of draws C until we accept. This C can be a function of the parameter $\backslash$kappa $.($b$)$ Write a function in R that employs the rejection algorithm with the candidate density that you have used in Part $($a$),$ in order to simulate n samples $($this will be given by the user$)$ from the distribution with the density in Equation $(0\mathrm{.}1)$ for $\backslash$kappa also given by the user $($they know that it should be a positive number and no further tests are required in your function$).$ The function should also return the total number of draws. $($c$)$ Test your function $($for n $=10,000)$ by generating a histogram of the output and by comparing the histogram with the target density. $($d$)$ Notice the total number of draws in Part $($c$)$ of the exercise. Does this seem to agree with the results from Part $($a$)$ of the exercise