TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2023W/HW12.3

Correlation 3

Let $X\sim {\mathcal {N}}(3,1)$ and $Y\sim {\mathcal {N}}(2,4)$. Plot 150 realizations of $(X,Y)$ ($y_{i}$ plotted against $x_{i}$, $i=1,\dots ,150$) for each of the values of their correlation $\mathbb {C} orr(X,Y)\in \{0,0.8,-0.5,-1\}$. To each plot add lines $\{(x,y)\in \mathbb {R} ^{2}\mid y=\beta _{0}+\beta _{1}x\}$ with $\beta _{0}$ and $\beta _{1}$ obtained according to Problem 2. Compute the empirical correlation $r$ of the

realizations and add it to your plots.

Dieses Beispiel ist als solved markiert. Ist dies falsch oder ungenau? Aktualisiere den Lösungsstatus (Details: Vorlage:Beispiel)

Lösungsvorschlag von Lessi

--Lessi 2024-02-07T13:04:11Z

# Let X ∼ N(3, 1) and Y ∼ N(2, 4). Plot 150 realizations of (X, Y ) (y_i plotted against x_i,
# i = 1, . . . , 150) for each of the values of their correlation Corr(X, Y ) ∈ {0, 0.8, −0.5, −1}. 

# To each plot add lines {(x, y) ∈ R^2 | y = β0 + β1x }
# with β0 and β1 obtained according to Problem 2. 

# Compute the empirical correlation r of the realizations and add it to your plots.

rm(list = ls())
par(mfrow=c(2,2))

n <- 150

mu_x <- 3
mu_y <- 2
sigma_x <- 1
sigma_y <- 2

x <- rnorm(n, mu_x, sigma_x)
y <- rnorm(n, mu_y, sigma_y)

plot_stuff <- function(rho) {
  beta_1 <- rho * sigma_y / sigma_x
  beta_0 <- mu_y - beta_1 * mu_x
  
  plot(x, y, main=paste(c("Correlation with rho=", rho), collapse = " "))
  
  abline(beta_0, beta_1, col='red')
  
  
  m_x <- mean(x)
  m_y <- mean(y)
  s_x <- sqrt(var(x))
  s_y <- sqrt(var(y))
  
  r <- cor(x, y)
  
  b_1 <- r * s_y / s_x
  b_0 <- m_y - b_1 * m_x
  
  abline(b_0, b_1, col='blue')
  mtext(paste(c("r=", r)), 3)
}

plot_stuff(0)
plot_stuff(0.8)
plot_stuff(-0.5)
plot_stuff(-1)