# Transforming a uniform-on-sphere random vector

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 $a,b,c$?

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