Uncorrelated random variables
For the last exercise, take a look at what happens when you have uncorrelated random variables. In other words, suppose that the random variables \(X\) and \(Y\) are bivariate normally distributed with correlation \(\rho_{XY} = 0\).
On the right, you find the code that contains the solution to the previous exercise.
This exercise is part of the course
Intro to Computational Finance with R
Exercise instructions
- Change the code to perform the same analysis with correlation \(\rho_{XY} = 0\) instead of \(\rho_{XY} = -0.9\).
Hands-on interactive exercise
Have a go at this exercise by completing this sample code.
# Standard deviations and correlation
sig_x <- 0.10
sig_y <- 0.05
rho_xy <- -0.9
sig_xy <- rho_xy * sig_x * sig_y
Sigma_xy <- matrix(c(sig_x ^ 2, sig_xy, sig_xy, sig_y ^ 2), nrow = 2, ncol = 2)
mu_x <- 0.05
mu_y <- 0.025
# Simulate 100 observations
set.seed(123)
xy_vals <- rmvnorm(100, mean = c(mu_x, mu_y), sigma = Sigma_xy)
head(xy_vals)
# Create plot
plot(xy_vals[, 1], xy_vals[, 2], pch = 16, cex = 2, col = "blue",
main = "Bivariate normal: rho = -0.9", xlab = "x", ylab = "y")
abline(h = mu_y, v = mu_x)
segments(x0 = 0, y0 = -1e10, x1 = 0, y1 = 0, col = "red")
segments(x0 = -1e10, y0 = 0, x1 = 0, y1 = 0, col = "red")
# Compute joint probability
pmvnorm(lower = c(-Inf, -Inf), upper = c(0, 0),
mean = c(mu_x, mu_y), sigma = Sigma_xy)