3AM stats quiz

PhilGoetz

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)

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)

Apologies; I get lost at step 3. Probably because I don't know stats that well. Hope this feedback helps.

I guess, what is meant by (I could be wrong though) is the joint probability distribution of X and Y which would look something like this but centered at the origin and there would be a surface where the height represents the probability density. (Also, ignore that the ellipse in this visualization is oddly rotated.) The contour lines are the ellipses you get by slicing the surface with a horizontal plane.

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