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While reading this answer, I clicked on the provided link to this Wikipedia page. The main article image shows the PDF of a 2D multivariate normally distributed system:

enter image description here

In the image, the $3\sigma$ ellipse is marked in green. Since the two variable coordinates x and y are normally (and thus symmetrically) distributed around the X and Y axes, I'd expect the ellipse's major and minor axes to be aligned with the X and Y axes. The image shows a slightly rotated ellipse.

Why is the ellipse rotated? Is it just a drawing error?

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ysap
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  • $X$ and $Y$ are not independent. – Hans Lundmark Jun 18 '14 at 15:10
  • @HansLundmark - where is this limitation mentioned in the article? Or, is it just your possible explanation to the rotation of the ellipe? – ysap Jun 18 '14 at 15:21
  • @ysap - Although it is not explicitly stated in the Wikipedia, it does appear that @HansLundmark is correct. The random variables are not independent.

    If you are familiar with R, then run the following:

    install.packages('mvtnorm') require(mvtnorm) #Sample from multivariate normal distribution of independent random variables obs <- rmvnorm(n = 10000, mean = c(0, 0), sigma = matrix(data = c(10, 0, 0, 1), nrow = 2)) plot(x = obs[, 1], y = obs[, 2], xlim = c(-15, 15), ylim = c(-15, 15))

     – Jubbles Jun 18 '14 at 23:25
  • @Jubbles - I am not familiar w/ R, however, I don't need that code to realize that dependent variables can rotate the ellipse. An extreme case is when x = y, so the ellipse is reduced to a 45deg straight line. – ysap Jun 19 '14 at 01:50
  • @HansLundmark - OK, I'll take it that this is the cause for the rotation in the image. If you make this an answer, I'll up-vote it. – ysap Jun 19 '14 at 01:50
  • All right, I wrote an answer (with a little explanation too). – Hans Lundmark Jun 19 '14 at 06:28

1 Answers1

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The random variables $X$ and $Y$ are not independent. As you can tell from the following line in the source code on the image page, there is a correlation between them:

CovMatrix = [1 0.6; 0.6 2];

(The off-diagonal entries in the covariance matrix are non-zero.)

Hans Lundmark
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  • Where do you see the source code?? – ysap Jun 19 '14 at 13:45
  • In the image description the OP mentions "Matlab code provided below.", but I see no code (nor in the 4 other pages where this image appear). – ysap Jun 19 '14 at 15:26
  • On http://en.wikipedia.org/wiki/File%3aMultivariateNormal.png, I just scroll down. It's between "Summary" and "Licensing". – Hans Lundmark Jun 19 '14 at 16:03
  • mm... I noticed that Wikipedia changed their interface some time ago. Now when I click on the image I get an "active" page, showing the image on a black background, with some dynamic info strip at the bottom. How did you get to that "static" page? – ysap Jun 19 '14 at 17:24
  • Looks like they call this thing "Media Viewer" – ysap Jun 19 '14 at 17:25
  • OK, I think I found it here: http://www.mediawiki.org/wiki/Multimedia/Media_Viewer/Help#How_can_I_bypass_Media_Viewer.3F – ysap Jun 19 '14 at 17:27
  • Every day you learn something new ;-) Thanks. – ysap Jun 19 '14 at 17:28
  • I just opened it in a new tab (like I usually do). And if I click on the link in my comment above, I get the static page. – Hans Lundmark Jun 19 '14 at 18:45