install.packages('https://cran.rstudio.com/bin/windows/contrib/3.5/MASS_1.2.zip')
## Installing package into 'C:/Users/dell/Documents/R/win-library/3.5'
## (as 'lib' is unspecified)
## inferring 'repos = NULL' from 'pkgs'
## Warning in download.file(p, destfile, method, mode = "wb", ...): cannot
## open URL 'https://cran.rstudio.com/bin/windows/contrib/3.5/MASS_1.2.zip':
## HTTP status was '404 Not Found'
install.packages('https://cran.rstudio.com/bin/windows/contrib/3.5/ggplot2_1.2.zip')
## Installing package into 'C:/Users/dell/Documents/R/win-library/3.5'
## (as 'lib' is unspecified)
## inferring 'repos = NULL' from 'pkgs'
## Warning in download.file(p, destfile, method, mode = "wb", ...):
## cannot open URL 'https://cran.rstudio.com/bin/windows/contrib/3.5/
## ggplot2_1.2.zip': HTTP status was '404 Not Found'
library(MASS)
## Warning: package 'MASS' was built under R version 3.5.2
library(ggplot2)
## Warning: package 'ggplot2' was built under R version 3.5.2
sigma=matrix(0,2,2)
sigma[1,1]=10
sigma[1,2]=0.9*sqrt(10)*sqrt(4)
sigma[2,1]=0.9*sqrt(10)*sqrt(4)
sigma[2,2]=4
mu=c(3,5)

La matriz de varianzas y covarianzas está dada por

sigma
##         [,1]   [,2]
## [1,] 10.0000 5.6921
## [2,]  5.6921 4.0000

El vector de esperanzas es

mu
## [1] 3 5

a) Simule 1000 observaciones de la distribución y genere un gráfico de dispersión.

x <- data.frame(mvrnorm(1000,mu,sigma))
ggplot(x, aes(X1, X2)) + geom_point(color = "deepskyblue",aes(colour="variables simuladas")) +
  labs(title= "Distribución Normal Multivariada", x = "X1", y="X2")

Al graficar las variables aleatorias simuladas podemos ver que se presenta una tendencia lineal positiva, esto se esperaba ya que tenemos un coeficiente de correlación cercano a 1.

b) Encuentre la media muestral y compárela con la media poblacional.

attach(x)
mu1 <- mean(X1) 
mu2 <- mean(X2) 
media_muestral <- c(mu1,mu2)

Para X1 tenemos una media poblacional de 3 mientras que la media muestral es 2.93 Para X2 tenemos una media poblacional de 5 mientras que la media muestral es 4.98

Para ambas variables nuestras medias muestrales fueron menores que las poblacionales.