Solve the following project in R$.$ Solve such that the code function will be able to calculate and plot histogram as output.

Project$$s description

This is a little fun project, and short, too. The project aims to define a class that allows the creation of new probability distributions based on existing ones. The solution needs to be a S$3$ class named randomVariate. The constructor of the class must accept two arguments. The first one, named distFormula, should be a formula, e$.$g$.,$ ~ x $+$ y$,$ describing the function of one or more random variables. The second argument, named defs, should be a list of functions used for sampling from given distributions with keys corresponding to symbols used in the formula. The created object of the class randomVariate should be a function used for sampling from the distribution defined by the formula. You should also write the print method specific to this class. The following examples should clarify the concept.

Example $1$

Let's start with a simple example. Let

be a standard normal random variable

$.$ The function used for sampling from this distribution in the R language is rnorm$().$ Suppose we want to define a new random variable

based on the random variable

$.$ Specifically, we want a function that allows for generating random numbers for this new random variable

$.$ We can define this new function using the constructor of the class randomVariate as follows.

### definition of the new variable

newDist $<-$ randomVariate$($

distFormula $=$ ~ x$^2,$

defs $=$ list$($

x $=$ rnorm

$)$

$)$

In the above snippet, we defined a new random variable

$.$ We used the constructor of the class randomVariate. The first argument of the constructor is a formula describing the function of the random variable

$.$ The second argument is a list defining the distribution of the random variable

$.$ Here, it is a standard normal distribution. The constructor creates a new object of the class randomVariate. We can print it out to see the mathematical definition of this new object.

newDist

output:

Formula

~x$^2$

Definitions

x

$------------$

function $($n$,$ mean $=0,$ sd $=1)$

$.$Call$($C$\_$rnorm, n$,$ mean, sd$)$

The object is printed in a specific way. First, a formula is printed, and then all used definitions. Here, there is only one definition for the symbol x$.$ In this particular case, the definition is a function rnorm$(),$ which is just a call to a compiled function C$\_$rnorm$().$

We can use the new object as a regular function to sample random numbers from the new distribution. The following snippet creates a random sample and plots a histogram based on the sample.

sample$1<-$ newDist$(10^5)$

hist$($sample$1,$ breaks $=$ "Scott", probability $=$ TRUE,

col $=$ rgb$(1,0,1,\mathrm{.}2),$ density $=20,$ angle $=-45)$

lines$($

density$($sample$1,$ adjust $=\mathrm{.}5),$

col $=$ rgb$(1,0,1,1))$

The above snippet generates the following graph. The new variable takes only non$-$negative values.