Generate a random variable which is uncorrelated with errors in r

Question

I generated an error in r

u1<-rnorm(n, mean=0, sd=1);

And I want to generate a random normal variable x ~ i.i.d.N(n, 0, 4) but this is not correlated with u1 i.e. corr(x, u1) is zero.

How can I, define this in R?

Answer 1

Generation of two uncorrelated variables:

n <- 1e3
u1 <- rnorm(n, mean = 0, sd = 1)
x <- rnorm(n, mean = 0, sd = 4)

Please note that estimated correlation coefficient will not be exactly 0, but it will be close to 0.

> cor(x, u1)
[1] -0.01885482

Answer 2

Any two random Gaussian variables will have a theoretical correlation of 0, as long as n is large enough.

set.seed(1)
n=1000000
u1<-rnorm(n, mean=0, sd=1)
x<-rnorm(n, mean=0, sd=4)

cor(x, u1)

Output:

0.0007637239