I want to answer this three questions and sub-questions manually (as a mathematically and statistically prove) then write it in R program (as R code)

(every questions and sub-questions have (mathematically or statistically) prove and R code)

(please noted I need for everything details and write the solutions, proves and codes step by step)

(Depend on your prove you write the R code)

These are the instructions to solve questions:

In question 1 sub-question a: use the moment generating function to prove it statistically then write R code.

In question 1 sub-question b: find the cumulative distribution function by prove it statistically then put k=2 then find the invers of cumulative distribution function by prove it mathematically (as F(x) = u and put x in one side and every things in the other side ) then write the code in R of the function.

In question 1 sub-question c: prove it statistically then write R code of function.

In question 1 sub-question d: draw two plots of histogram of sub-question a and c.

In question 2 sub-question a: prove it statistically then write R code of function.

In question 2 sub-question b: prove it statistically then write R code of function and plot it.

In question 3 method A: use pox muller transformation to prove method A mathematically and write R function of method A.

In question 3 method B: prove it statistically for every (i, ii, iii, iv) then write R code of function for every (i, ii, iii, iv).

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