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Consider the 3-D real random vector $(X_1,X_2,X_3)$ which is uniformly distributed on the surface of a unit sphere. What can be told about the distribution of $(aX_1,bX_2,cX_3)$, where $a,b,c,$ are non-zero and non-identical real constants? Is it true to say that they are distributed uniformly on the surface of an ellipsoid with the corresponding parameters $a,b,c$?

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    $\begingroup$ Considering the 2D case and contemplating the situation where $b\gg a$ ought to provide good intuition as to why the answer is no. $\endgroup$ – whuber Dec 19 '13 at 16:45
  • $\begingroup$ @whuber: I see, so uniformity is not preserved. Now, can we say anything about the distribution of the scaled vector? $\endgroup$ – Ebrahim Dec 19 '13 at 18:38
  • $\begingroup$ Of course! One of the simplest and most elegant ways to describe it is that the back-transformed values (on the sphere) are uniform :-). $\endgroup$ – whuber Dec 19 '13 at 19:18

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