R 问题与蒙特卡罗模拟

Question

How I could solve this exercise:

Let U come from a uniform (0,1) distribution and Z come from a standard normal distribution. Let X = pZ + (1-p)U. Estimate p when X has a variance of 0.4.

Maybe I should install.packages('polynom')?

p is equal to 0,62 because I´ve already calculated it analitically, but I don´t know how to solve it with R using random random numbers generators

Answer 1

I will not solve for you the exercice but give you the clue on how to solve it

You need to do as the exercice does:

1. Simulate U[0,1]
2. Simulate N(0,1)

Put them together using a value for p

For a given p , you can define a function:

set.seed(234)
computeX <- function(p){

  u <- runif(1000)
  z <- rnorm(1000)

  X = p*z + (1-p)*u
  return(X)
}

var(computeX(0.2))
[1] 0.08924463

Now I let you find the routine to get p that is needed for your exercice.

No need for any external library

Answer 2

Here is an example

fobj <- function(p)  abs(var(p*rnorm(1e5) + (1-p)*runif(1e5))-0.4)
pval <- optimise(fobj,c(0,1))$minimum

where

• first define your objective function fobj , ie, distance between variance of X and 0.4
• then apply optimise minimize the fobj

Result

> pval
[1] 0.6223968