Use R code!!!!!!!

We wish to generate realizations from the following pdf using the Accept$-$Reject method:

$(|)=K(-x^2)$;$>0,-\sqrt[\phantom{2}]{}x\sqrt[\phantom{2}]{}$

$K$ is the normalizing constant so that it is a proper pdf$.$ The value of $K$ is not difficult to find and is

$K=\frac{3}{4{}^{\frac{3}{2}}}.($Also note that $(|)=0$ for or $x>\sqrt[\phantom{2}]{}.)$

The mean of the above distribution is clearly equal to $0$ as it is symmetric about $0.$ So the first step is

to find the variance of the above distribution, ${}^2=V[x]=E[x^2].$

Recall that in the A$-$R method, a candidate generating distribution $g(x)$ is needed along with a

constant $M$ such that $\frac{f(x)}{g(x)}M$ for all $x.$

The candidate generating distribution that I want you to use is the Normal $(0,{}^2)$ distribution where

${}^2$ is the variance calculated above. The value of $M$ will not be provided by the user, so you have

two choices to determine its value automatically:

You can determine its value by maximizing $f\frac{x}{g}(x)$ mathematically $($i$.$e$.$ taking derivative,

setting equal to $0,$ etc.$)$ to find $x_m=argma{x}_{x}f\frac{x}{g}(x)$ and letting $M=f\frac{x_m}{g}(x_m).$

You can determine its value by maximizing $f\frac{x}{g}(x)$ empirically by searching for the value of

$x_m$ numerically. I would suggest a simple grid search, but if your background includes

knowledge of numerical optimization, then feel free to utilize it$.$

With the preliminaries out of the way, on to the task at hand. Write a function called "myrdist" to

generate $n$ realizations from $(|)$ above using the Accept$-$Reject method.

The function's name must be "myrdist".

It must take the following arguments $($named as such$)$:

$n-$ The sample size; i$.$e$.$ the number of realizations to generate.

Include error checking to be sure that $n$ is a numeric scalar and greater than or

equal to $1.$

If it is not an integer, then coerce into the nearest integer less than the value

given.

No default value.

beta $-$ This is the value of $.$

Include error checking to be sure that beta is a numeric scalar and greater than

$0.$

The default value is $1.$

It must return a list with the following components:

The vector of $n$ realizations, named "realizations".

The value of $M$ used, named $"M".$

The average number of iterations required per realization $($i$.$e$.$ the number of candidate

$x^{\text{'}}$ s generated divided by $n),$ named "avg,iterations".

We wish to generate realizations from the following pdf using the Accept$-$Reject method:

$(|)=K(-x^2)$;$>0,-\sqrt[\phantom{2}]{}x\sqrt[\phantom{2}]{}$

$K$ is the normalizing constant so that it is a proper pdf$.$ The value of $K$ is not difficult to find and is

$K=\frac{3}{4{}^{\frac{3}{2}}}.($Also note that $(|)=0$ for or $x>\sqrt[\phantom{2}]{}.)$

The mean of the above distribution is clearly equal to $0$ as it is symmetric about $0.$ So the first step is

to find the variance of the above distribution, ${}^2=V[x]=E[x^2].$

Recall that in the A$-$R method, a candidate generating distribution $g(x)$ is needed along with a

constant $M$ such that $\frac{f(x)}{g(x)}M$ for all $x.$

The candidate generating distribution that I want you to use is the Normal $(0,{}^2)$ distribution where

${}^2$ is the variance calculated above. The value of $M$ will not be provided by the user, so you have

two choices to determine its value automatically:

You can determine its value by maximizing $f\frac{x}{g}(x)$ mathematically $($i$.$e$.$ taking derivative,

setting equal to $0,$ etc.$)$ to find $x_m=argma{x}_{x}f\frac{x}{g}(x)$ and letting

Instructions $-$ i should be able to copy and paste the R code and execute without having to track down aby bugs $.$ Dont use global variables