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title: ’’ author: “Hagia” date: ‘2023-06-13’ output: html_document

library(mvtnorm) 
library(ggplot2)

library(matlib)

## Standard deviation
sigma <-matrix(c(4,2,2,3),ncol=2,nrow=2)
## Mean
mu <-c(1,2)

n <- 1000

set.seed(123)

x <-rmvnorm(n=n,mean=mu,sigma=sigma)

d <-data.frame(x)

p2 <-ggplot(d,aes(x=X1,y=X2))+
geom_point(alpha =.5)+
geom_density_2d()
p2

y <-x-mu

E <-eigen(sigma)
E$vectors

##            [,1]       [,2]
## [1,] -0.7882054  0.6154122
## [2,] -0.6154122 -0.7882054

y <-y%*%t(inv(E$vectors))

dd <-data.frame(y)

p3 <-ggplot(dd,aes(x=X1,y=X2))+
geom_point(alpha =.5)+
geom_density_2d()
p3