Because I'm tired and easily amused.

- Consider two independent, normally-distributed variables X and Y, each with a mean of 0.
- X has a standard deviation of 4.
- Y has a standard deviation of 1.

- Consider the probability distribution function (in two dimensions) for <X, Y>.
- Imagine contour lines on the PDF connecting points of equal probability density. These would be a series of concentric ellipses.
- Draw the contour line that is tangent to the line X = 4.
- What values of Y does it pass through when X = 0?
- Post your answer.
- Run this R code to plot the contours:

library(mvtnorm)

bivn <- mvrnorm(100000, mu = c(0, 0), Sigma = matrix(c(4, 0, 0, 1), 2, 2))

bivn.kde <- kde2d(bivn[,1], bivn[,2], n = 100)

image(bivn.kde); contour(bivn.kde, add = T)