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I'm currently revising for an exam and I don't understand one of the questions. Given are the following expectation / mean vectors and covariance matrices:

A) $\mu_1 = \begin{pmatrix}0 \\ 0 \end{pmatrix}, \Sigma_1 = \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}$

B) $\mu_2 = \begin{pmatrix}3 \\ 2 \end{pmatrix}, \Sigma_2 = \begin{pmatrix} 4 & 0 \\ 0 & 1\end{pmatrix}$

C) $\mu_3 = \begin{pmatrix}3 \\ 2 \end{pmatrix}, \Sigma_3 = \frac{1}{\sqrt{10}}\begin{pmatrix} 1 & -3 \\ 3 & 1\end{pmatrix}\begin{pmatrix} 4 & 0 \\ 0 & 1\end{pmatrix}\frac{1}{\sqrt{10}}\begin{pmatrix} 1 & -3 \\ 3 & 1\end{pmatrix}^T$

where the 3rd covariance matrix is in the form $UDU^T$. We must then use this to draw the scatter diagram, as shown below enter image description here

I don't understand this at all. Why is the radius for $A$ 3 and not 1? Why are the $B$ and $C$ drawn like they are? The only part I understand is that the center of the ellipse/circle is the expectation/mean vector and that the eigenvectors should be used (but not how)...

John
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user5368737
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  • About $98.9\%$ of your points will be within the circle in A. About $39.3\%$ would be inside a circle radius $1$ – Henry May 29 '17 at 21:28
  • @Henry how do you know that though? I don't understand! – user5368737 May 29 '17 at 22:45
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    I used R code for a $\chi^2$-distribution: `pchisq(3^2, df=2); pchisq(1^2, df=2)` and checked with simulation – Henry May 29 '17 at 22:49
  • @Henry yes but this is an exam so I'm expected to be able to do it on paper. – user5368737 May 29 '17 at 23:12
  • The radii are arbitrary. From the fact that a radius of $3$ was used to depict the density in case (a), you immediately conclude the corresponding radii in cases (b) and (c) must be $3$ times $\sqrt{1}$ and $\sqrt{4}=2$; that is, they must be $3$ and $6$. No more calculation than that is needed. We cannot tell you where the original value of $3$ came from: that's some convention used by whoever provided these answers. If you would like to see some carefully drawn images of the same sort, with explanations of their connections, then see my post at https://stats.stackexchange.com/a/71303/919. – whuber May 30 '17 at 14:02
  • @whuber I understand why it's a radius of 3 now (3 * standard deviation). I still don't understand c) though, how am I supposed to draw an ellipse from UDU^T? Or the other way around (UDU^T from a given ellipse)? – user5368737 May 31 '17 at 16:58
  • $U$ is a rotation matrix: as you can confirm by multiplying, its first column tells you where $(1,0)$ goes and its second column where $(0,1)$ goes. That's all you need to figure out how to draw the ellipse (and I assume it's the entire point of part (c) of the question). – whuber May 31 '17 at 17:45

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